Hilbert Spaces of Analytical Functions

•

-1 e aces 0 •

une Ions

Preface

The workshop entitled Hilbert Spaces of Analytic Functions was held at the Centre de recherches matMmatiques (CRM), Montreal, from 8 to 12 December 2008. Even though this event was not a part of the CRM thematic year, 62 mathematicians attended the workshop. They formed a blend of researchers with a common inteIest in spaces of analytic functions, but seen from many different angles.

lhlbert spaces of analytic functions are currently a very active field of complex analyslS. The Hardy space H2 is the most senior member of this family. Its relatlVes, such as the Bergman space AP, the Dirichlet space V, the de BrangesRovnyak spaces 1£ b), and various spaces of entire functions, have been extensively studIed by prominent mathematicians since the beginning of the last century. These spaces ha\ been explOlted in different fields of mathematics and also in physics and engmeering. For example, de Branges used them to solve the Bieberbach conjecture, and Zames, a late professor of McGill University, applied them to construct hlS th ry f HOC control. But there are still many open problems, old and new, which attract a wid spectrum of mathematicians.

In thlS nference, 38 speakers talked about Hilbert spaces of analytic functions. In five days a wi e vanety of applications were discussed. It was a lively atmosphere m which many mutual research projects were designed.

xi

J avad Masbreghi Thomas Ransford

Kristian Seip

The production of this volume was supported in part by the Fonds Quebecois de la Recherche sur la. Nature et les Technologies (FQRNT) and the Natural Sciences aDd Engineering Research Council of Ca.nada (NSERC).

2000 Mathematics Subject Classification. Primary 46E20, 46E22, 47B32, 31C25.

Library of Congress Cataloging-In-Publication Data

Hilbert spaces of analytic functions / Javad Masbreghi, Thomas Ransford, Kristian SeIP, echtora. p. cm. - (CRM proceedings & lecture notes; v. 51)

Includes bibliographical references. ISBN 978-0-8218-4879-1 (alk. paper) 1. Hilbert space-Congresses. 2. Analytic functions-Congresses. 3. P tentJal theory

(Mathematics)-Congresses. I. Mashreghi, Javad. 11. Ransford, Thomas. 111. 8eJ.p, Knsban., 1962-

QA322.4.H55 2010 515'.733---dc22

2010003337

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10 9 8 7 6 5 4 S 2 1 15 14 13 12 11 10

Volume 51

Centre de Recherches Mathematlques Montreal

•

1

J avad Mashreghi Thomas Ransford Kristian Seip Editors

aces 0

The Centre de Recherches Mathematiques (CRM) of the Universire de Montreal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Universite de Montreal, the Province of Quebec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the lnstitut des Sciences Mathematiques (ISM) of Montreal. The CRM may be reached on the Web at www.clm.math.ca.

American Mathematical Society Providence. Rhode Island USA

List of Participants

List of Speakers

Contents

Canonical de Branges Rovnyak Model 'Iransfer-Function Realization for Multivariable Schur-Class Functions

.. Vll

• IX

• Xl

Joseph A. Ball and Vladimir Bolotnikov 1

Two Variations on the Drury Arveson Space NUXJla Arcozzi, Richard Rochberg, and Eric Swayer 41

The Norm of a Truncated Toeplitz Operator Stephan Ramon Garcia and William T. Ross 59

Approximation in Weighted Hardy Spaces for the Unit Disc Andre Bo vin and Changzhong Zhu 65

Some Remarks on the Toeplitz Corona Problem Ronald Douglas and Jaydeb Sarkar 81

on the BOllndary in Spaces of Holomorphic Functions on the Unit Disc

Emmanuel Fricain and Andreas Hartmann 91

The Search for Singula.rities of Solutions to the Dirichlet Problem: Recent Developments

Dmitry Khavinson and Eric Lundberg 121

Invariant Subspaces of the Dirichlet Space Omar El-Fallah, Karim Kellay, and Thomas Ransford 133

Arguments of Zero Sets in the Dirichlet Space Javad Mashreghi, Thomas Ransford, and Mahmood Shabankhah 143

Questions on Volterra Operators Jaroslov Zemanek 149

Nonhomogeneous Div-Curl Decompositions for Local Hardy Spaces on a Domain

Der-Chen Chang, Galia Dafni, and Hong Yue 153

On the Bohr Radius for Simply Connected Plane Domains Richard Fournier and Stephan Ruscheweyh 16E

vi CONTENTS

Completeness of the System {/(AnZ)} in L�[nJ Andre Boivin and Changzhong Zhu

A Formula for the Logarithmic Derivative and Its Applications Javad Mashreghi

Composition Operators on the Minimal Mobius Invariant Space Hasi Wulan and Chengji Xiong

Whether Regularity is Local for the Generalized Dirichlet Problem Paul M. Gauthier

173

197

203

211

List of Participants

Evgueni AbakOllmov Universite Paris-Est

Fatma Z. Abdessameud Universite Saad Dahlab Blida

Jim Agler UC San Diego

Thyeb Ai"ssiou McGill University

Nadya Askanpour University of Western Ontario

Hirbod Assa. Universite de Montreal

Amoah G. AtingabllDor Khomananj College

Joseph A. Ball VIrginia Tech

Ferenc Balogh Concordia University

Laurent Baratcbart INRlA Sophia Antipolis-Mediterranee

Andre Boivin University of Western Ontario

Alexander Borichev Universite de Provence

Abdellatif Bourbim Universite Laval

Marcus Carlsson Purdue University

Nicolas Chevrot Universite Laval

Yemon Choi Universite Laval

Joseph A. Cima UNC Chapel Hill

Constantin Costara Universitatea Ovidius

Galla Dafni Concordia University

Ronald G. Douglas Texas A&M University

Omar EI-Fallah Universite Mohammed V

Yasser Farhat Universite Laval

Tatyana Foth University of Western Ontario

Richard Fournier Universite de Montreal

Emmanuel Fricain Universite Claude Bernard Lyon 1

Frederic Gaunard Universite Bordeaux 1

Paul M. Gauthier Universite de Montreal

Kohur Gowrisankaran McGill University

Dominique Guillot Universite Laval

Andreas Hartmann Universite Bordeaux 1

vii

vIII

Tarik Jari Universite Laval

Dmitry Khavinson University of South Florida

Daniela Kraus Universitii.t Wiirzburg

Alex Louis University of Swaziland

Yura Lyubarskii NTNU

Mostafa Mache Universite Laval

Davood Malekzadeh Universite Laval

Jordi Marzo NTNU

Javad Mashreghi Universite Laval

Tesfa Mengestie NTNU

Putinar Mihai UC Santa Barbara

Joules Nahas UC Santa Barbara

Mostafa N asri Universite Laval

Joaquim Ortega-Cerda. Universitat de Barcelona

Vladimir Peller Michigan State University

Mihai Putinar UC Santa Barbara

Quentin Rajon Universite Laval

Thomas J. Ransford U niversite Laval

Richard Rochberg

PARTICIPANTS

William T. Ross University of llichmond

Stephan Ruscheweyh Universitat Wiirzburg

Sergey Sadov Memorial University of'iewfoundland

Kristian Seip NTNU

Alexis Selezneff Universite Laval

Mahmood Shabankhah Universite Laval

Richard Spjut UC Santa Barbara

Qingyun Wang Washington Univers ty in St. Lows

Hasi Wulan Shantou Um erslty

Nicolas Y ung Leeds University

HongYue ConcordIa University

Jaroslav Zemanek Polish Academy of SCIences

Nina Zorboska University of Manitoba

Washington University in St. Louis

List of Speakers

Evgueni Abakllmov Universite Paris-Est

Fatma Z. Abdessameud Universite Saad Da.hlab Blida

Jim Agler UC San Diego

Joseph A. Ball Virginia Tech

Ferenc Balogh Concordia University

Laurent Baratchart INRIA Sophia Antipolis - Meditarranee

Andre Boivin University of Western Ontario

Alexander Borichev Universite de Provence

Abdelatif Bourbim Universite Laval

Nicolas Chevrot Universite Laval

Joseph Cima UNC Chapel Hi11

Constantin Costara Universitatea Ovidius

Ga.1ia Dafni Concordia University

Ronald G. Douglas Texas A&M University

Omar EI-Fallah Universit.e Mohammed V

Tatyana Foth University of Western Ontario

Richard Fournier Universite de Montreal

Emmanuel Fricain Universite Lyon 1

Paul Gauthier Universite de Montreal

Kohur GowriSanka.ran McGill University

Andreas Hartmann Universite Bordeaux 1

Dmitry Khavinson University of South Florida

Danieal Kraus Universitat Wiirzburg

Yura Lyubarskii NTNU

Jordi Marzo NTNU

ix

Joules Nahas UC Santa Barbara

Joaquim Ortega-Cerda Universitat de Barcelona

Vladimir Peller Michigan State University

Mihai Putinar UC Santa Barbara

Thomas Ransford Universite Laval

x

Richard Rochberg Washington University in St. Lo

William T. Ross University of Richmond

Stephan Ruscheweyh Universitiit Wiirzburg

Mahmood Shabankhah Universite Laval

Hassi Wulan Shantou University

Nicolas Young University of Leeds

Hong Yue Concordia University

J aroslav Zemanek Polish Academy of Science

C~nt .. de Recherchee Math~matlqu"" CRM Proceedingl and J ecture Notea Volume &1. 2010

Canonical de Branges - Rovnyak Model Transfer-Function Realization for Multivariable Schur-Class Functions

Joseph A. Ball and Vladimir Bolotnikov

ABSTRACT. Associated with any Schur-c111S8 function S(z) (i.e., a contractive holomorphic function on the the unit disk) is the de Branges-Rovnyak kernel Ks{~, C) = [1-S{z)S(C)·]/(l- z() and the de Branges- Rovnyak reproducing kernel Hilbert spsce 1i(Ks). This space plays a prominent role in system theory as a canonical-model state space for a transfer-function realization of a given Schur-class function. There has been recent work extending tbe notion of Schur-cJ1IRS function to several multivariable settings. We bere ma.ke explicit to what extent the role of de Branges-Rovnyak spaces as the canonical-model state space for transfer-function realizations of Schur-class functions extends to multivariable settings.

1. Introduction

Let U and Y be two Hilbert spaces and let L(U, Y) be the space of all bounded linear operators between U a.nd y. The operator-valued version of the classical Schur class S(U,Y) is defined to be the set of all holomorphic, contractive L(U, Y)valued functions on 1Il. The following equivalent characterizations of the Schur class are well known. Here we use the notation H2 for the Hardy space over the unit disk and ~ = H2 ® X for the Hardy space with values in the auxiliary Hilbert space X.

Theorem 1.1. Let S: 1Il -+ L(U, Y) be given. Then the following are equivalent:

(1) (a) S E S(U, Y), i.e., 8 is holomorphic on 1Il with 118(z)11 ~ 1 for all z E 1Il.

(b) The opemtor Ms: fez) I-t 8(z)f(z) of multiplication by 8 defines a contmction opemtor from H~ to H~.

(c) S satisfies the von Neumann inequality: 118(T)1I ~ 1 for any strictly contmctive opemtor T on a Hilbert space 11., where 8(T) is defined

2000 Mathellwzbcs Subject ClasBifico.tion. 47A57. Key words and phrasea. Operator-valued functions, Schur multiplier, canonical functional

model. reploducing kernel Hilbert space. This is the final {olm of the paper.

@20l0 American Mathematical Society

1

2

(1.1)

J. A. BALL AND V. BOLOTNIKOV

by 00 'Xl

SeT) = L Sn ® rn E C(U ® 1/., Y ® 1/.) if S(z) = LSnz'l. n=O

(2) The associated kernel

Ks(z, () = Iy - S(z)~«t 1-z(

n=O

is positive on JD) x JD), i.e., there exists an operator-valued functwn H. D-+ C(X, Y) for some auxiliary Hilbert space X so that

(1.2) Ks(z, () = H(z)H«t·

(3) There is an auxiliary Hilbert space X and a umtary conned ng operator

so that S (z) can be expressed as

(1.3) S(z) = D + zO(I - zA)-l B.

(4) S(z) has a realization as in (1.3) where the connect ng opemtorU u one of (i) isometric, (ii) coisometnc, or (iii controch e.

We note that the equivalence of any of (la), (lb, 1c with 2 and can be gleaned, e.g., from Lemma V.3.2, Proposition 1.8.3, Proposition V.S.1 and Theorem V.3.1 in [26]. As for condition (4), it is trivial to see that 3 imphe:s 1 and then it is easy to verify directly that (4) implies (la . Alternatl,-ely, one can use Lemma 5.1 from Ando's notes [6] to see directly that 4iii implies .fu see Remark 2.2 below).

The reproducing kernel Hilbert space 1/.(Ks) with the de Branges-Rorn ilk kernel Ks(z, () is the classical de Branges - Rovnyak reproducmg kernel Hrlberl space associated with the Schur-class function S which has been much studied O\-ef

the years, both as an object in itself and as a tool for other types of applications see [6,11-13,16-18,20,21,24,27,28,31]). The special role of the de Branges-Ro\'Il'8k space in connection with the transfer-function realization for Schur-class functIons is illustrated in the following theorem; this form of the results appears at Iea:.1

implicitly in the work of de Branges Rovnyak [20,21].

Theorem 1.2. Suppose that the function S is in the Schur class S(U, y and e 1-£(Ks) be the associated de Branges Rovnyak space. Define operators A,B,C,D by

A: fez) 1-+ fez) - f(O) , z

C: fez) 1-+ f(O),

B: u I-? S(z) - S(O) u, z

D: u I-? S(O)u.

Then the operator-block matrix U = l ~ ~] has the following properties: (1) U defines a coisometry from 1/.(Ks) e U to 1/.(Ks) e y. (2) (C,I\) is an observable pair, i.e.,

CAn f ... 0 for all n = 0,1,2, ... """'=> J = 0 as an element of1-£(Ks).

(3) We recO'Uer S(z) as S(z) = D ;.. zO(I _ .tA) lB.

TRANSFER-FUNCTION REALIZATION

(4) If [~: g: 1 : X Ell U -+ X ffi Y is another colligation matrix with properhes (1), (2), (3) above (with X in place of 1£(Ks)), then there is a unitary operator U: 1£(Ks) -+ X so that

U 0 A B A' B' U 0 -o Iy C D - C' D' 0 Iu'

It is easily seen from characterization (la) in Theorem 1.1 that

(1.4) S E S(U,Y) < ;- S E S(Y,U) where S(z) := S(i)*.

Hence for a given Schur-class function S there is also associated a dual de BrangesRovnyak space ?-leKs) with reproducing kernel Ks(z, () = [1 -8(i)* S(())/(l-z(). The space ?-leKs) plays the same role for isometric realizations of 8 as ?-l(Ks) plays for coisometric realizations, as illustrated in the next theorem; this theorem is just the dual version of Theorem 1.2 upon application of the transformation (1.4).

Theorem 1.3. Suppose that the function S is in the Schur class S(U,Y) and let ?-leKs) be the associated dual de Branges-Rovnyak space. Define operators Ai, Bd, Cd, Dd by

A.i: g(z) 1-+ zg(z) - S(i)* yeO), Bd: u 1-+ (I - S(i)* S(O) )u,

Cd: g(z) 1-+ yeO), Dd: u 1-+ S(O)u,

whel e y 0) I.S the unique vector in Y such that

-(O)} ( ) S(i)* - S(O)* 9 ,y y = 9 z , Y z

for all y E y.

Then the operator-block matrix U d = l~: g~ 1 has the following properties:

1) Ud defines an isometry from ?-leKs) Ell U to 'tl.(Ks) Ell y, 2 (Ai,Bd) is a controllable pair, i.e., Vn>oRanA'dBd = 'tl.(Ks), where V -stands for the closed linear span. 3 We recover S(z) as S(z) = Dd + zCd(I - ZAd)-lBd' 4) If [~' g',1 : X Ell U -+ X Ell Y is another colligation matrix with properties

(1 , (2), (3) above (with X in place of 1£(Ks )), then there is a unitary operator U: 'tl. (K s) -+ X so that

U 0 Ad Bd A' B' U 0 -o Iy Cd Dd - C' D' 0 Iu' ~

In addition to the kernels Ks and Ks , there is a positive kernel Ks which combines two and is defined as follows:

(1.5) ~ Ks(z,() K(z, () = S(z)-S(q

%-c ~

S(%)-S(~) %-, Ks(z, ()

I-S(z)8JQ·, _ 1-%~ - S z o_s C •

%-I -s(.;:) 8m .

1-ol( ol-

It tW"IIS out that K is also a positive kernel on Jl)) x Jl)) and the associated reproducing ~

kernel Hilbert space ?-l(Ks) is the canonical functional-model state space for unitary realizations of 8 , as summarized in the following theorem. This result also appears at least implicitly in the work of de Branges and Rovnyak [20,21J and more explicitly

~

the paper of de Branges and Sbulman [22J, where the two-component space ?-leKs) associated with the Schur-clfl13l3 function S is denoted as D(S); see also [11J for an explanation of the connections with the Sz.-Nagy-Foias model space.

4 J. A. BALL AND V. BOLOTNIKOV

Theorem 1.4. Suppose that the function S is in the Schur class S U,y) aM let K(z,() be the positive kernel on JI} given by (1.5). Define operators A,B,c fi by

A' r f(z)1 t-+ r [fez) - f(0)1I z 1 . Lg(z) Lzg(z) - S(z)f(O) , B~ . f([S(z) - S(O 1 z u1

. u I-t l (I - S(z)* SOu

8: r f(z)1 t-+ f(O) Lg(z) , D: u I-t S(O)u.

Then the operator-block matrix U = [~ ~ 1 satisfies the following:

(1) U defines a unitary operator /rom 1£ (K s) G3 U onto 1£ K s G3 Y· (2) U is a closely connected operator colligation, i.e.,

V {Ran..4.n B, Ran..4.*nC*} = 1£(Ks). n~O

(3) We recover S(z) as S(z) = D + zC(I - z..4.)-1 B. (4) If [~: g:] : X G3 U -+ X G3 Y is any other opemtor colligatwn sahsfym9

conditions (1), (2), (3) above (with X in place of 1£ Ks , then there IS

unitary operator U: 1£(Ks) -+ X so that

rU 01 fA Bl fA' B'1 ru 01 L 0 Iy l C D J = L 0' D' lo Iu'

Our goal in this article is to present multivariable analogues of Theorem 1.2. The multivariable settings which we shall discuss are (1) the unit ball Bel in C' and the associated Schur class of contractive multipliers between vector-valued Drury Arveson spaces lI.u (kd) and 11. y (kd), (2) the polydisk with the associated Schur class taken to be the class of contractive operator-valued functions on Dei which satlsfy a von Neumann inequality, and (3) a more general setting where the underlying domain is characterized via a polynomial-matrix defining function and the Schur class is defined by the appropriate analogue of the von Neumann inequality. In the'le multivariable settings, the analogues of Theorem 1.1 have already been set down at length elsewhere (see [3,15,23] for the ball case, [1,2,14] for the polydisk case, and [4,5,9] for the case of domains with polynomial-matrix defining function-see [8] for a survey). Our emphasis here is to make explicit how Theorem 1.2 can be extended to these multivariable settings. While the reproducing kernel spaces themselves appear in a straightforward fashion, the canonical model operators on these spaces are more muddled: in the coisometric case, while the analogues of the output operator 0 and the feedthrough operator D are tied down, there is no canonical choice of the analogue of the state operator A and the input operator B: A and B are required to solve certain types of Gleason problems; we refer to [25] and [30, Section 6.6] for some perspective on the Gleason problems in general. The Gleason property can be formulated also in terms of the adjoint operators A' and B': the actions of the adjoint operators are prescribed on a certain canonically prescribed proper subspace of the whole state space. From this latter formulation, one can see that the Gleason problem, although a.t first sight a.ppearing to be rather complicated, always has solutions. Also, the adjoint of the colligation matrix, rather than being isometric, is required only to be isometric on a certain subspace of the whole space X EB y. With these adjustments, Theorem 1.2 goes through in the

TRANSFER-FUNCTION REALIZATION 5

three settings. Most of these results a.ppear in [10] for the ba.ll case and in more implicit form in [14] for the polydisk case, although not in the precise formulation presented here. The parallel results for the third setting are presented here for the first time. We plan to discuss multivariable analogs of Theorems 1.3 and 1.4 in a. future publication.

The paper is organized as follows. After the present Introduction, Section 2 lays out the results for the ball case, Section 3 for the polydisk case, and Section 4 for the case of domains with polynomial-matrix defining function. At the end of Section 4 we indicate how the results of Sections 2 and 3 can be recovered as special

of the general formalism in Section 4.

2. de Branges-Rovnyak kernel associated with a Schur multiplier on the Drury - Arveson space

A natural extension of the Szego kernel is the Drury - Arveson kernel

1 1 kd(Z,() = =: ( .

1-Z1(1-"'-Zd(d 1- Z,()Cd

The kernel kd Z, () is positive on Bd x Bd where

Bd = {z = (Zl' ... , Zd) E Cd: (z, z) = \Zl\2 + ... + \Zd\2 < I}

is the unit ball in Cd, and the associated reproducing kernel Hilbert space ll(kd) is called the Drury-Arveson space. For X any auxiliary Hilbert space, we use the shorthand notation llX(kd) for the space ll(kd) ® X of vector-valued DruryArveson-space functions. A holomorphic operator-valued function S: Bd -+ LeU, y) is to be a Drury Arveson space multiplier if the multiplication operator Ms: / z t--+ S Z fez) defines a bounded operator from llU(kd) to lly(kd)' In case in addition Ms defines a contraction operator (I\Msl\op 5 1), we say that S is in the Schur-multiplier class Sd(U, Y). Then the following theorem is the ana.logue of Theorem 1.1 for this setting; this result appears in [10,15,23). The alert reader will notice that there is no analogue of condition (la) in Theorem 1.1 in the following theorem.

Theorem 2.1. Let S: Bd -+ l.(U, Y) be given. Then the following are equivalent:

2.1)

1) (b S E Sd(U, Y), i.e., the operator Ms of multiplication by 8 defines a contraction operator from llU(kd) into lly(kd)'

c) S satisfies the von Neumann inequality: \\8(T)1\ 5 1 for any commutative operator d-tuple T = (Tl' ... , Td) of operators on a Hilbert space /C such that the operator-block row matrix [Tl . .. Td] defines a stnct contraction operator from /Cd into /C, where

SeT) = L Sn®-rn E l.(U®ll,Y®ll) nEZi

if 8(z) = L 8n zn.

nEZd +

Here we use the standard multivariable notation:

z'" = Z~1 .,. Z~d and ~ = 'If1 ... T:;d if n = (nl," ., nd) E Zi. (2) The associated kemel

(2.2) K ( 1") = 1y - 8(z)8«>* s z,... 1 _ (z, ()


is positive on B x B, i. e., there exists an opemtor-valued function H . B.t -+ C(X, Y) for some a'I.IXiliary Hilbert space X so that Ks(z, <) = H(z H < •

(3) There is an a'I.IXiliary Hilbert space X and a unitary connectmg operator

(2.3) u- [~ ~l = l~ ~1' [~1-+ l~ so that 8(z) can be expressed as

(2.4) 8(z) = D + 0(1 - Zrow(Z)A)-l Zrow(z)B,

where we have set

Zrow(z) = (zlIx ... zd1x).

(4) 8(z) has a realization as in (2.4) where the connecting opemtorU as an one of (i) isometric, (ii) coisometric, or (iii) contmctwe.

Remark 2.2. Statement (4iii) concerning contractive realizations is not mentioned in [15] but is discussed in [10,23]. The approach in [231 is to show that for 8 of the form (2.4) with U = [~ g] contractive the inequality S T) S 1 h ds for any commutative operator d-tuple T = (Tlo ••• , Td with (Tl ..• Td1 < I, Le, one verifies (4iii) :::::} (Ie).

The idea of the second approach in [10] is to embed the contraction U = {~g into a coisometry [~ ~] = (~ g g~] with associated transfer function of the form.

S(z) = (8(z) 81 (z)] equal to an extension of S(z) with a larger input space. From

the coisometry property of [~~] one sees that Ks(z,w) = 0 1 -Zrow z A -1 1-

A· Zrow«)-IC·, i.e., S meets condition (2) for the Schur-class Sd U eUhY With H(z) = C(l - Zrow(z)A)-l. From the equivalence (lb) ~ (2 , it is easy no" to read off that 8 E Sd(U, y).

A third approach worked out for the classical case but extendable to multr variable settings appears in Ando's notes [6, Lemma 5.11. Given a contractiw colligation U = [~g], one can keep the input and output spaces the same but

enlarge the state space to construct a coisometric colligation U = [~~ 1 having the same transfer function, namely:

r Qll Q12 0 0

.. ] n=m'

• 0 0 0 I 0 ... A= 0 0 0 0 I ,

c= (0 Q21 Q22 0 o ... ], D=D where

Q = [~i~ ~~~] = (I - UU*)1/2.

In this way one gets a direct proof of (4iii) ==> (4ii).

For colligations U of the form (2.3), it turns out that a somewhat weaker notior. of coisometry is more useful than simply requiring that U be coisometric.

TRANSFER·FUNCTION REALIZATION 7

Definition 2.3. The operator-block matrix U of the form (2.3) is weakly coisometric if the restriction of U" to the subspace

(2.5)

is isometric.

Du-:= V (EBcI

\lEY

Zrow«)*(l - A* Zrow«)*)-lO"y C Xd Y Y

It turllS out that the weak-coisometry property of the colligation (2.3) is exactly what is needed to guarantee the decomposition

(2.6) Ks(z,() = O(l - Zrow(z)A)(l - A"Zrow«)*)-lC*

of the de Branges-Rovnyak kernel Ks associated with S of the form (2.4) (see Proposition 1.5 in [10]).

2.1. Weakly coisometric canonical functional-model colligations. As the kernel Ks given by (2.2) is positive on B x B, we can associate a reproducing kernel Hilbert space 1l(Ks) just as in the classical case, where now the elements of 1l Ks are holomorphic Y-valued functions on Bd. In the classical case, as we

from Theorem 1.2, there are canonically defined operators A, B, C, D so that the operator-block matrix U = [~~l is coisometric from ll(Ks) ffiU to ll(Ks) ffi Y and yields the essentially unique observable, coisometric realization for S E S U,y. For the present Drury-Arveson space setting, a similar result holds, but the operators A, B in the colligation matrix U are not completely uniquely determined. To expJajn the result, we say that the operator A: ll(Ks) -+ 1l(KS)d s I es the Glea-c;on problem for 1l(Ks) if the identity

d

2.7 / z) - f(0) = L zk(Afh(z) holds for all f E 1l(Ks) , k=l

(Alh (,,)

where we write Af) z) = • • •

E 1l(KS)d. We say that the operator B: U-+ (AIM")

1l Ks d solves the 1l(Ks)-Gleason problem for S if the identity

d

2.8 S z)u - S(O)u = LZk(Bu)k(Z) holds for all u E U. k=l

Solutions of such Gleason problems are easily characterized in terms of adjoint operators.

Proposition 2.4. The operator A: ll(Ks) -+ 1l(KS)d solves the Gleason problem for ll(Ks) (2.7) if and only if A*: 1l(KS)d -+ ll(Ks) has the following action all. special kernel functions:

2.9) A*: Zrow«)* Ksh ()y ~ Ks(o, ()y - Ksh O)y for all (E Bd, Y E y.

The operator B: U -+ 1l(KS)d solves the ll(Ks)-Gleason problem for S (2.8) if and only if B*: 1l(KS)d -+ U has the following action on special kernel functions:

(2.10) B*: Zrow«)* Ks("()Y t-+ S«)*y - S(O)"y for all (E Bd, Y E y.


PROOF. By the reproducing kernel property, we have for I E l£(Ks),

(I(Z) - f(O),y}y = (f, Ks("z)y - Ks(·,O)Y}1£(Ks)·

On the other hand,

(t, zk(Af)k(Z), y))I = t«Af)k' ZkKS(" Z)Y}1£(Ks

= (AI, Zrow(Z)* Ks(-, Z)Y}1£ Ks d

= (f,A*Zraw(z)*Ks("z)y 1£(Ks

and since the two latter equalities hold for all I E 1l(Ks), z E Bd and Y E :v the equivalence of (2.7) and (2.9) follows. Equivalence of (2.8) and 2.10 fol1~ similarly from the computation:

(t, zk(Bu)k(z), y))I = t, «Bu)k, ZkKS(" z)y 1£ Ks

= (Bu, Zrow(z)* Ks(-, z Y 1£ Ks

= (u, B* Zraw(z)* Ks("z)y u·

Let us introduce the notation

(2.11) v = V Zrow«)" Ks(" ()y. t;Eud

yE)I

o

Definition 2.5. Given S E Sd(U, y), we shall say that the block-operoto! matrix U = [~ ~ 1 is a canonical functional-model colligation for S if

(1) U is contractive and the state space equals 1l(Ks}. (2) A: l£(Ks) -+l£(Ks)d solves the Gleason problem for 1l Ks (2.7. (3) B: U -+l£(Ks)d solves the 1l(Ks)-Gleason problem for S 2.8). (4) The operators C: 1l(Ks) -7- Y and D: U -7- Yare given by

(2.12) C: I(z) 1-+ 1(0), D: U 1-+ S(O)u.

Remark 2.6. It is useful to have the formulas for the adjoints C" : Y -+11. Ks and D: Y -+U:

(2.13) C*: y 1-+ Ks(-, O)y D* : y 1-+ S(O)*y

which are equivalent to (2.12). The formula for D· is obvious while the formula. for C· follows from equalities

(f,C·yhi(Ks) = (C/,y}Y = (f(O),y}y = (f,Ks(·,O)Y}1£(Ks)

holding for every f E l£(Ks) and y E y.

Theorem 2.7. There exists a canonical functional-model realization for etlery 5 E Sd(U,y).

PROOF. Let 5 be in Sd(U, Y) and let 1l(Ks) be the associated de Branges Rovnyak space. Equality (2.2) can be rearranged as

d

L z,(,Ks(z, () + 1)/ = Ks(z, () + S(z)S«)* ;-1

TRANSFER· FUNCTION REALIZATION

which in turn, can be written in the inner-product form 88 the identity

(2.14) Zraw«()* Ks(', ()y

y I Zrow(z)* Ks(" Z)y'

y' 'H(K s )ci$)!

- K s (', ()y Ks(', Z)y' -

9

S«(ty , S(Zty' 'H(Ks)$U

holding for every y, y' E Y and (, z E Bd. The latter identity tells us that the linear map

(2.15)

extends to the isometry from Vy = V EB Y C 1i(KS)d EB Y (where V is given in (2.11)) onto

.,., - V Ks(', ()y (K) U "-y - S«()*y C 1i s EB •

CESci liE)!

Extend V to a contraction U .. : 1i(KS)d EB Y ~ 1i(Ks) EBU. Thus,

2.16) U· - A* C* . Zrow«()* Ks(" ()y ~ Ks(', ()y - B* D* . y S«()*y .

Comparing the top and the bottom components in (2.16) gives

2.17 A* Zrow«()* K s (" ()y + C*y = Ks(" ()y,

2.18 B* Zrow«()* K s (" ()y + D*y = S«()*y.

Solving (2.17

2.19)

for Ksc-, ()y gives

Ks("()Y = (1 - A* Zrow«()*)-lC"y.

Substituting this into (2.18) then gives

2.20 B* Zrow«()*(1 - A* Zraw«(t)-lC*y + D*y = S«(ty.

By taking adjoints and using the fact that ( E Ba and y E Yare arbitrary, we may then conclude that U is a contractive realization for S. It remains to show that U meets the requirements (2) - (4) in Definition 2.5. To this end, we let ( = 0 in 2.17) and (2.18) to get

(2.21) C*y = Ks("O)y and D*y = S(O)*y.

Substituting (2.21) back into (2.17) and (2.18), we get equalities (2.9) and (2.10) which are equivalent (by Proposition 2.4) to A and B solving the Gleason problems (2.7) and (2.8), respectively. By Remark 2.6, equalities (2.21) are equivalent to (2.12). 0

Remark 2.8. A consequence of the isometry property of V in (2.15) is that fot IIIlIlas (2.9) and (2.10) extend by linearity and continuity to give rise to uniquely detelll.ined well-defined linear operators Ai> and BD from V to 1i(Ks) and U, respectively. In this way we that the existence problem for operators A solving the Gleason problem is settled: A: l£(Ks) ~ l£(Ks)d solves the Gleason problem forl£(Ks) (2.7) if and. only if A" is an extension to all of1i(Ks)d of the opemtor Ai,: V ~ 1i(Ks) uniquely detennined by the formula (2.9). Similarly, the opemtor B: U ~ 1i(KS)d is a solution of the 1i(Ks)-Gleason problem for S (2.8) if and only


if the operator B*: ll(KS)d -+ U is an extension to all of ll(KS)d of the 0Jl"TIIt9I' BD: V -+ U uniquely determined from the fonnula (2.10).

The following result is essentially contained in {IO}. For the ball setting, we 1l&!

the following definition of observability: given an operator pair C, A WIth 0 tp operator C: X -+ Y and with A: X -+ X d , we say that C, A is obsenable If C(I - Zrow(z)A)-lx = 0 for all z in a neighborhood of 0 in Cd implies that % = in X. Equivalently, this means that

V (I - A* Zrow(zt)-lC·y = X "EL!. yEY

for some neighborhood 6. of 0 in Cd.

Theorem 2.9. Let S be a Schur-class multzpher in Sd U,)J and suppose tha: U = [~ ~ 1 is any canonical functional-model colhgatlon for S. Then:

(1) U is weakly coisometric. (2) The pair (C, A) is observable. (3) We recover S(z) as S(z) = D + C(l - Zrow(Z A -lZrow Z B (4) If U' = [~: g: 1 : X E9 U -+ Xd E9 Y is any other II gahon matn:r en

joying properties (1), (2), (3), then there is a canon cal functw'oolcolligation U = [~ ~ 1 : ll(Ks) E9 U -+ l£(Ks d E9 Y so that U IS umt equivalent to U /, i. e., there is a umta1"Y operator U: X -+ 1£ Ks so that

[~ ~1 [~ ~Y1 = [EB~Ol U ~1l~ ~1· PROOF. Since U is a canonical functional-model colligation for S, the opm

tors A and B solve the Gleason problems (2.7) and (2.8 , respectively. By Prop<>sition 2.4, this is equivalent to identities

A* Zrow«)* Ks("()Y = Ks("()Y - Ks ·,0 Y

B* Zrow«)* Ks(" ()y = S«)*y - S(O)*y.

Besides, C and D are defined by formulas (2.12). Substituting their adjoints from (2.13) into the two latter equalities we arrive at (2.17) and (2.18. As v.e ha -e seen, equalities (2.17) and (2.18) imply (2.19) and (2.20). Equality 2.20 PIO\'l'S

statement (3). Equality (2.19) gives

(2.22) V (I - A* Zrow«)*C·y = V Ks(" ()y = ll(Ks). {EBd {EBd

yEY !,lEY

Thus the identity C(I - Zrow(z)A)-l/ == a leads to (J, (I - A* Zrow(Z)*)-lC'Y "" 0

for every z e IRd and y e Y; this together with equality (2.22) implies f == 0, and it follows that the pair (C, A) is observable.

On the other hand, equalities (2.17) and (2.18) are equivalent to (2.16). Substituting (2.19) into (2.16) and in (2.14) (for z = ( and y = y') gives

U. rZrow«)*(1 - A* Zrow«)*)-lC*Y) = r(I - A* Zrow«)*)-lC*y1 l y l S«)*y

and


respectively. The two latter equalities tell us that U· is isometric on the 'Du. (see (2.5» and therefore U is weakly coisometric. For the proof of part (4) we refer to [10, Theorem 3.4]. 0

Definition 2.5 does not require U to be a realization for S: representation (2.4) is automatic once the operators A, B, 0 and D are of the required form. We can look at this from a different point of view as follows. Let us say that A: 1I.(Ks) -+ 1I.(KS)d is a contractive solution of the Gleason problem (2.7) if in addition to (2.7), inequality

d

(2.23) L\I(Af)kl\~(Ks) ::; \Ifll~(Ks) -lIf(O)II~ k=l

holds for every f E 1i(Ks). It is readily seen tha.t inequality (2.23) can be equiva.lently written in operator form as

where the operator 0: 1i(Ks) -+ Y is given in (2.12). It therefore follows from Definition 2 5 that for every canonical functional-model colligation U = l ~ g] for S, the operator A is a contractive solution of the Gleason problem (2.7). The following theorem provides a converse to this statement.

Theorem 2.10. Let S E Sd(U,Y) be given and let us assume that 0, Dare !I' en by fonllulas (2.12). Then

1 For every contractive solution A of the Gleason problem (2.7) for 11. (Ks) , there exists an operator B : U -+ 11. (K s) such that U = [~ g] is contrach e and S zs reahzed as in (2.4).

2 E ery such B solves the 1I.(Ks)-Gleason problem (2.8) so that U is a canonical functional-model colligation.

PROOF. Smce A solves the Gleason problem (2.7) and since 0 is defined as in 2.12 , we conclude as in the proof of Theorem 2.9 that identity (2.17) holds which

is equivalent to (2.19). On account of (2.19), it is readily seen that (2.18) and (2.20) are equivalent. But (2.20) is just the adjoint form of (2.4) whereas (2.18) coincides with 2.10 since D = S(O)) which in turn, is equivalent to (2.8) by Proposition 2.4. Thus, it remains to show that there exists an operator B*: 1I.(Ks) -+ U completely detew.ined on the subspace 'D C 1I.(Ks) by formula (2.10) and such that U· = [~: ~: 1 is contractive. This demonstration can be found in [10, Theorem 2.4]. 0

3. de Branges - Rovnyak kernels associated with a Schur - Agler-class function on the polydisk

Here we introduce a generalized Schur class, called Schur - Agler class, associated with the unit polydisk

Jl)d = {z = (Zl, ••• ,Zd) E Cd: Iz,.1 < 1 for k = 1, ... ,d}.

We define the Schur-Agler class SAd(U,Y) to consist of holomorphic functions S: nd -+ £(U, Y) such that I\S(T) II ::; 1 for any collection of d commuting opera.tors T = (Tlo"" Td) on a Hilbert space JC with liT,. II < 1 for each k = 1, ... , d where the operator SeT) is defined 88 in (2.1).


The following result appears in [1,2,14] and is another multivariable analogup of Theorem 1.1. The reader will notice that analogues of both (la) and lb fr m Theorem 1.1 are missing in this theorem.

Theorem 3.1. Let S be a c'(U, Y) -valued function defined on Del. The following statements are equivalent:

(3.1)

(1) (c) S belongs to the class SAd(U, Y), i.e., S satisfies the von NeufTl.4Tlrt inequality \IS(Tl. ... , Td)1I :5 1 for any commutative d-tuple T ::: (T1, ••• , Td) of strict contraction operators on an a1.lXtl,ary Ht1hert space /C.

(2) There exist positive kernels KI, ... ,Kd:]Dd x]Dd -+ c'(Y) such that/or every z = (Zl. ... , Zd) and ( = «(1. •.• , (d) in]Dd,

d

ly - S(z)S«()* = 2)1- z.(.)K.(z,(). k=l

(3) There exist Hilbert spaces Xl,"" Xd and a unitary connectmg operator U of the structured form

[A B1 lA~l . . . A~d ~11l~11 l~11

(3.2) U = 0 D = A:dl A:dd

~d : ;d -+ ;d

(3.3)

(3.4)

0 1 Od D U Y

so that S(z) can be realized in the form

S(z) = D + 0(1 - Zdiag(Z)A) -1 Zdiag(z)B for all z E Dd

where we have set

(4) There exist Hilbert spaces Xl, ... I Xd and a contractive connecnng operator U of the form (3.2) so that S(z) can be realized in the form (3.3)

Remark 3.2. Although statement (4) in Theorem 3.1 concerning contracti\1l realizations does not appear in [1,2,14], its equivalence to statements (1)-(3) can be seen by anyone of the three approaches mentioned in Remark 2.2.

Similar to the notion introduced above for the unit-ball case, there is a notion of weak coisometry for the polydisk setting as follows.

Definition 3.3. The operator-block matrix U of the form (3.2) is weakly cois/)" metric if the restriction of U· to the subspace

(3.5) Vu.:= V [Zdlag«()*(1 - A~Zdiag«()·)-lO*Y] c [~d] <eDd

IIeY

is isometric.


When U is given by (3.2) and S(z) is given by (3.3), it is immediate that we ha.ve the equality

[C(1 - Zdia.g(z)A)-lZdia.g(Z) I} U = [C(1 - Zdiag(z)A)-l S(z)}.

From this it is easy to verify the following general identity:

(3.6) 1- S(z)S()" = C(1 - Z(z)A)-l(1 - Z(z)Z()")(I - A"Z(t)-1C"

where here we set Z(z) = Zdia.g(Z) for short. It is readily seen from (3.6) that the wea.k-coisometry property of the colligation (3.2) is exactly what is needed to guarantee the representation

(3.7) 1- S(z)S()"

= C(1 - Zdia.g(z)A)-1(1 - Zdiag(Z)Zdia.g()")(I - A" Zdiag(),,)-1C".

Note that the representation (3.7) has the form (3.1) if we take

3.8) Kk(z, () = C(1 - Zdiag(z)A)-1 Px" (I - A" Zdiag(),,)-1C"

for k = 1, ..• , d, where Px" is the orthogonal projection of X := E9~=1 Xi onto Xk •

3.1. Weakly coisometric canonical functional-model colligations. Let us say that a collection of positive kernels {K1(z,(), ... ,Kd(Z,()} for which the decomposition 3.1) holds is an Agler decomposition for S. In view of (3.7), we see that a' 3.3) for S arising from a weakly coisometric colligation matrix U 3.2 determines a particular Agler decomposition, namely that given by (3.8).

Our next goal is to find a canonical weakly coisometric realization for S compatible with the given Agler decomposition. Toward this goal we make the following definitions.

Suppose that we are given a Schur-Agler class function S E SAd(U,y) to-gether with an Agler decomposition {K1(z, (), ... , Kd(Z, ()) for S. We set

K(z, () = K 1(z, () + ... + Kd(Z, ().

Then lK is also a positive kernel on Jl))d and the associated reproducing kernel Hilbert space 1l K) can be characterized as

d

1l(K) = L Ii : Ii E 11.(Ki) for i = 1, ... ,d i=1

with norm given by

i=1

where 8: E9~-11l(K,) -+ 11.(K) is the linear map defined by

(3.9) d

where J = EB Ii :=:: i=1

• • • •


It is clear that kers = {J E EB~=lll(Ki) : /l(z) + ... + Jd(Z) == O}. If we let

lKl(Z,()] 11'(z, () ;= : '

Kd(z,()

(3.10)

we observe that by the reproducing kernel property, d

(3.11) (I, 11'(., ()Y)E9t~ll£(K,) = L)Ii, K i (·, ()Y}l£(K,) i=1

so that

(3.12) s·: 1K(·, ()y --+ 11'(., ()y.

Furthermore, d

(kers).L = V 11'(.,()y c ffi ll(Kk ).

(EDd k=1 yE)I

We next introduce the subspace

(3.13) v = V Zdiag«)"''ll'(·,()y (ED" yE)I

of EB~=11l(Kk) and observe that its orthogonal complement can be described as

d d d V.L = {I = ~ Ii E ~ll(K,) : ~Z.i,(z) = o}-

In addition, the straightforward computation d

1111'(·,()YIla,~=l1t(Kk) = 2:(Kk«,()Y,Y}Y = (K«,()y,y}y = 1I1K(·,()y ~(K) k=1

combined with (3.12) shows that s* is an isometry, Le., that s is a coisometry· We remark that all the items introduced so far are uniquely determined from decomposition (3.1).

Given an operator-block matrix A = [A'Jlt..,=1 acting on EB~=1 ll(K,), we;:} that A solves the structured Gleason problem lor the kernel collection {KlI"" if the identity

d

(3.14) !I(z) + ... + Id(Z) - [/1(0) + ... + ideO)] = :L z.(Af),(z) ,=1

d d m d (Af).(Z) holds for all 1= EB.=1 I. E EB i =1 ll(Ki), where we write (AI)(z) == 1;17>==1

E EB~_lll(K.). Note that (3.14) can be written more compactly as d d

(3.15) (sl)(z) - (sl)(O) = Lz,(A •• f)(z) for all I E E9ll(K/r)' , 1 " 1

TRANSFER-FUNCTION REALIZATION 1~

where s is given in (3.9) and where

d

(3.16) A •• = (A.l ... A'd)! Et)1l(Kk) -41l(K.) (i = 1, ... , d) k=l

so that d d d

(3.17) A=Et)A •• =lA'Jl~'J=l: Et)1l(Kk)-4Et)1l(Ki). k=l i=l

We say that the operator B: U -4 Ea:-l1l(Kk) solves the structured ll(Kk)Gleason problem for S if the identity

(3.18) S(z)u - S(O)u = zl(Buh(z) + ... + zd(Bu)d(Z) holds for all u E U.

The following is the parallel to Proposition 2.4 for the polydisk setting.

Proposition 3.4. The operator A: Eat=l1l(Ki ) -4 Eat=l1l(Ki) solves the stru tured Gleason problem (3.15) if and only if the adjoint operator A* has the folloWtng a t\on on SpecIal kernel functions:

(lKl·,()y K1(·,()y Kl("O)y • • •

• • •

- • • •

for all ( E lIJ)d and y E y.

The opemtor B. U -4 Ea~=lll(Ki) solves the structured Gleason problem (3.18) f S fad on y if the adjo~nt operator B*: Eat=lll(Ki ) -4 U has the following act\ n n spectal kernel junctions:

( Kl - ()y • • •

(dKd("()Y

t-+ S«()*y - S(O)*y for all ( E lIJ)d and y E y.

PROOF. Making use of notation (3.10) and (3.4) we can write the definitions of A * BDd B* more compactly as

3.19

3.20

A* Zdiag«()*11'(·, ()y = 11'(., ()y - 11'(., O)y,

B* Zdiag«()*11'(·, ()y = S«()*y - S(O)"y.

By calculation (3.11),

1,1'(·, ()y -1'(" O)Y)E9~=l 1i(K,) = (sf) «() - (sl)(O), y}y.

On the other hand, it follows by the reproducing kernel property that

I, A* Zd,ag(Z)*1'(·, Z)Y)E9~ 11i(K,) = (Zdiag(z)AI, 11'(., Z)Y)E9t~l1i(K,) d

= "ElZdiag(z)A!1i(Z),y ~=l )I

d

= "E zilAf).(z), Y

and the two latter equalities show that (3.15) holds if and only if (3.19) is in force for every y E y. Equivalence of (3.18) and (3.20) is verified quite similarly. 0


The following definition of a canonical functional-model colligation is the ~ logue of Definition 2.5 for the polydisk setting.

Definition 3.5. Given S E SAd(U,Y), we shall say that the block-()peratr.t matrix U = [~g 1 of the form (3.2) is a canonical functional-model coli gatlon associated with the Agler decomposition (3.1) for S if

(1) U is contractive and the state space equals ffi~=11l(K.). (2) A: EB1=11l(Ki) -4 ffi1=11l(Ki) solves the structured Gleason problem

(3.15). (3) B: U -4 EB1=11l(Ki) solves the structured Gleason problem 3.18 for S

(4) The operators C: EBt=11l(Ki) ~ Y and D: U ~ Y are given by

(3.21) C: f(z) ~ (81)(0), D: 11. ~ S(O)1I..

Remark 3.6. For C and D defined in (3.21), the adjoint operators are gi.\"1!Ii by

(3.22) C*: Y ~ 'll'(',O)y D*: y ~ S(O)*y.

The formula for D* is obvious while the formula for C* follows from equalities

(I, C*Y)(Bt=l1i(K;) = (C f, y)y = ((sl)(O), y}y = (f, 'll' ·,O)y (B =11l K

holding for every f E EB1=11l(Ki).

Theorem 3.7. Let S be a given function in the Schur-Agler class SA.:! U,y) and suppose that we are given an Agler decomposition (3.1) fOT S. Then there e:nsts a canonical functional-model colligation associated with {K 1, .. " Kd}.

PROOF. Let us represent a given Agler decomposition (3.1) in the inner product form as

d

l)(iKi(" ()y, ZiKi(', z)y')1i(K,) + (y, y')y ~l d

= L (Ki (·, ()y, K.(., z)y'ht(K,> + (S«()*y, S(z)'!I~u, i=1

or equivalently, as

(3.23) ([Zdiag(();'ll'("()y], [Zdiag(z);!(.,z)y']) (EB~=l 'H(K,) )GlY

1['ll'("()Y) ['ll'("Z)y']) = \ S«()*y , S(z)*y' «(B~=l1l(K,))$U

where l' is given in (3.10). The latter identity implies that the map

(3.24) V: [Zdias(()*'ll'("()y] ~ ['ll'(" ()y] y S«()*y

extends by linearity and continuity to an isometry from Vv = V EB Y (8. subspace of (EBt=lll(Ki )) EB Y-(3.13) for definition of V) onto

'R- = V ['ll'("()Y] C [ffi~_lll(Ki)] V S«()*y U'

~EDd,IlEY


-Let us extend V to a. contraction U· :

(3.25)

Computing the top and bottom components in (3.25) gives

(3.26)

(3.27)

A· Zdie.g «).'11' (., ()y + C·y = '11'(" ()y,

B· Zdie.g«)·'11'(·, ()y + D·y = S«ry.

11

Letting ( = 0 in the latter equalities yields (3.22) which means that C and D are ofthe requisite form (3.21). By substituting (3.22) into (3.26) and (3.27), we arrive at (3.19) and (3.20) which in turn are equivalent to (3.15) and (3.18), respectively. Thus, U meets all the requirements of Definition 3.5. 0

We have the following parallel of Remark 2.8 for the polydisk setting.

Remark 3.B. As a consequence of the isometric property of the operator -V 3.24) introduced in the proof of Theorem 3.7, formulas (3.19) and (3.20) can be extended. by linearity and continuity to define uniquely determined operators AD: 'D -+ ffi~=lll(K.) and BD: 'D -+ U where the subspace 'D of ffi~=lll(Ki) is defined. in (3.13). In view of Proposition 3.4, we see that the existence question

is then settled.: any opemtor A: ffi~=l1-£(Ki) -+ ffi~=lll(Ki) such that A· is an

of AD from 'D to all offfi~=lll(Ki) is a solution of the structured Gleason

3.15) and any opemtor B: U -+ ffi1=11-£(Ki ) so that B· is an extension of the operator BD: 'D -+ U is a solution of the structured Gleason problem (3.18) forS.

In the polydisk setting we use the following definition of observability: given an operator A on ffi~=l.:ti and an operator C: ffi~=lXi -+ y, the pair (C,A) wi11 be called obsenJable if equalities C(1 - Zdiag(z)A)-l Px,x = 0 for all z in a neighborhood of the origin and for all i = 1, ... , d forces x = 0 in ffi~=l Xi. The latter is equivalent to the equality

3.28 V Px.(1 - A· Zdiag(Z)·)-lC·y = Xi for i = 1, ... , d zEa,I/E)/

for some neighborhood f:l. of the origin in Cd. The following theorem is the analogue of Theorem 1.2 for the polydisk setting; portions of this theorem appear already in {14, Section 3.3.1].

Theorem 3.9. Let S be a function in the Schur - Agler class SAd(U, Y) with a gtven Agler decomposition {K1 , •••• Kd} for S and let us suppose that

(3.29)

u a canonical functional-model colligation associated with this decomposition. Then:

(I) U is weakly coisometric. (2) The pair (C. A) is obsenJable in the sense of (3.28). (3) We recover S(z) as S(z) = D + 0(1 - Zdiag(z)A)-l Zdle.g(z)B.


(4) Iff; = (~~1 : (EB~=l x.) ffiU ~ (EB~=l x.) ffi Y is any other co tg~ matrix enjoying properties (1), (2), (3) above, then there l8 Q ca~ functional-mo~l colligation U = [~Z 1 as in (3.29) whICh l8 ~ equivalent to U in the sense that there are unitary operators U . X. -+ 1i(Ki) 80 that

(3.30) [A B) r EB~=l Ui 0) = r EB~=l U. 01 r ~ ~l C D l 0 IJ1 l 0 lu lc Dr

PROOF. Let U = [~ Z 1 be a canonical functional-model realization of S a&sI).

ciated with a fixed Agler decomposition (3.1). Then combining equalities 319 (3.20) (equivalent to the given (3.15) and (3.18) by Proposition 3.4 and also formulas (3.22) (equivalent to the given (3.21)) leads us to

(3.31) i(·, ()y = (1 - A" Zdiag«)*)-l'lI'(·,O)y = (1 - A" ZdIag ( " -lc;*y

and

(3.32) 8«()*y = 8(0)*y + B" Zdiag«)*'lI'(·, ()y = D"y + B" ZdIag ( "T . C Substituting (3.31) into (3.32) and taking into account that y E Y is arbItrary get

(3.33) 8«)* = 8(0)" + B" Zdiag«)"(1 - A" Zd. g ( " -le"

which proves part (3) of the theorem. Also we have from 3.31 and 3.1 ,

V P'H.(K;)(l-A"Zdiag«)")-lC"y= V P'HK 1'.,(y <End (End yEJI yEll

= V K, ., ( y = 11. K, (ElDd

yEll

and the pair (C, A) is observable in the sense of (3.28). On the other hand, equahoe5 (3.19), (3.20) are equivalent to (3.25). Substituting (3.31) into 3.25 and mto identity (3.23) (for z = ( and y = y') gives

U. [Z«)" (1 - A" Z«)")-lC"y) = r(1 - A" Z«)")-1C* y1 y l 8«)"y

and

\\lZ«()" (I - A~Z«)")-lC"Y1 \\ = \\ ((1 - A"~~~~~-lC"Y1 \' respectively. The two latter equalities show that U* is isometric on the space Vu(see (3.5)) and therefore U is weakly coisometric.

To prove part (4), let us assume that

(3.34) 8(z) = 8(0) + C(1 - Zdiag(z)A)-l Zdiag(z)B

is a weakly coisometric realization of 8 with the state space EB~=l X, and such tha.t the pair (C,A) is observable in the sense of (3.28). Then 8 admits en }.gler decomposition (3.1) with kernels K, defined as in (3.8):

K,(z, () = C(l - Zdiag(z)A)-l px• (1 - A" Zdlag«)*) lC"


for i = 1, ... , d. Let ll(Ki) be the associated reproducing kernel Hilbert spaces and let :r. : X, -+ X = ffi~=l X, be the inclusion maps

~: Xi -+ 0 e ... e 0 e Xi e 0 e ... e o. - -Since the pair (C, A) is observable, the operators U,: X, -+ ll(K.) given by

- - 1 (3.35) U,: Xi -+ C(l - Zdia.g(z)A)- Z.Xi

are unitary. Let US define A E .c(ffi~=lll(K,)) and B E .c(U, ffit=11l(K.)) by

(3.36) and ,=1 ,=1 .=1

In more detail: A = IA"l~"=1 where

. - - -1 - - -1 -(3.37) A". C(l - Zdia.g(z)A) I,xj -+ C(l - Zdiag(z)A) ~AijXj.

Define the operators A,. as in (3.16) and similarly the operators Ai. for i = 1, ... , d.

Take the generic element f of ffi~=11l(Ki) and X E X in the form

d

3.38 f z) = EBO(lx - Zdie.g(z)A)-IIjXj, 3=1

d

X=EBXjEX. j=1

By 3 37 • we have

3.39

d

• • •

= LAij(O(l - Zdiag(z)A)-1IjXj) ;=1

d

= L 0(1 - Zdiag(z)A)-I~AijXj 3=1 - .. 1 -= C(l - Zdie.g(z)A)- ~Ai.X,

For f and :z: as in (3.38), we have

d d

sf) z) = L O(lx - Zdiag(z)A)-IIjxj = O(lx - Zdiag(z)A)-1 ~ IjX; ,=1 j=1

= O(1x - Zdiag(z)A)-l x

which together with (3.39) gives

(sf)(z) - (sf)(O) = 0(1- Zdia.g(z)A)-l x - Ox - - 1 .. = C(l - Zdie.g(z)A)- Zdiag(z)Ax

d d

= E zi .0(1- Zdie.g(z)A)-lI,Aj.x :::: L Zj • (Aj.f)(z), 1=1 j~l


which means (since f is the generic element of EB~=lll(K.)) that the operatms Ale,'" ,AM satisfy identity (3.15). Furthermore, on account of (3.38), (3.35 and (3.34),

d d

LZi(BuMz) = LZiC(l - Zdiag(z)A)-lI,Biu i=1 i=1

d

= C(I - Zdiag(z)A)-l I>.I;B.u .. =1

- - 1 = C(l - Zdiag(z)A)- Zdiag(z)Bu = S(z)u - SOu

and thus, B solves the Gleason problem (3.18) for S. On the other hand, for an:r of the form (3.38), for operators Ui defined in (3.35), and for the operator C defined

on EBt=1'H.(Ki ) by formula (3.21), we have

(

d d d d

C ~ Ui) X = tr(UiXi) (0) = ~ C(l - Zdiag(O)A)-lI,x. = C~:r.z. = C:r

and thus C(EBt=l Ui) = C. The latter equality together with definitions 336 implies (3.30). Thus the realization U = [~ Z 1 is unitarily equivalent to the ong·

inal realization U = (~~ 1 via the unitary operator EB~=l U.. This realization IS

a canonical functional-model realization associated with the Agler decompootion {Kb ... , Kd} of S since all the requirements in Definition 3.5 are met. 0

We conclude this section with a theorem parallel to Theorem 2.9. In analogy with the ball setting, we say that the operator A on EB~=lll(K. is a controctn-e solution of the structured Gleason problem for the kernel collection {K1, ••• ,Kd} if in addition to identity (3.15) the inequality

d

IIAfll~t=l1i(Ki) := LIIAi.fll~(K.) ~ IIfll~~=l 1i(K.) - (sl)(O) ~ i=l

holds for every function f e EB:=lll(Ki) or equivalently, the pair (C,A) is contractive:

A'"A+C'"C ~ 1,

where C: EBt=l 'H.(Ki) -4 Y is the operator given in (3.21). By Definition 3.5, {or every canonical functional-model colligation U = (~ Z J associated with a gh-en Agler decomposition of S, the operator A is a contractive solution of the structured Glea.son problem (3.15).

Theo1:em 3.10. Let t~.l) be a fixed Agler decomposition of a given juncnOfi S E SAd(U,y) and let C and D be defined as in (3.21). Then

(1) For every contractive solution A of the structured Gleason problem (3.15),

there is an operator B = EB B,: U -+ EB:=lll(K.) such that U = l~ Zl is a canonical functional-model colligation for S.

(2) Every such B solves the Gleason problem (3.18) for S.

PROOF. We start the proof with two preliminary steps.


Step 1. Let A of the form (3.17) solve the Gleason problem (3.15). Then

d

(3.40) 0(1 - Zdia.g(z)A)-1 f = (81)(Z) Z E Jl)d j f E E91i(Ki) ,=1

where sand C are defined in (3.9) and (3.21), respectively.

PROOF OF STEP 1. To show that identity (3.15) is equivalent to (3.40) we take A in the form (3.17) and define the operators

Ale 0 0 0 A2• • ~ • • • (3.41) Ale = A2e = Ad. = • , • , • , ... , • • 0 • •

0 0 Ad.

d

SO that A •• : E9~=1 1i(K,) -+ E9~=11i(Ki) and A = 2: Ai.' On account of (3.41) .=1

and due to the block structure (3.4) of Zdia.g(Z) we have

-1 .... ,. -1 (1 - Zdia.g(z)A) = (1 - Zl Ah - '" - ZdAd.)

Applying the operator e(l - Zdiag(z)A)-1 to an arbitrary f E E9~=11i(Ki) and making use of formula (3.21) for C, we get

3.42 e 1 - Zdtag(z)A)-l f 00

= e ~)zlAh + ... + zdAd.)k f k=O

d d

= (81)(0) + LZi(sAi.I)(O) + 2: zizj(sAi.Aj.I)(O) + .... • =1 i,j=l

On the other ba,nd, by writing (3.15) in the form

d

(sl)(z) = (sl)(O) + L: zi(sAi.I)(Z) ;'=1

and iterating the latter fOlmula for each f E E9~=11i(Ki)' we get

(3.43) (s/)(z) d d

= (81)(0) + L Z;1 (SA31 .1)(0) + L Zh (sAh.Ail.f)(O) + .,. ]1=1 h=l

d

+ L Zjh [(sAj/o. '" A3~.Ajl.I)(0) + ... J... . 310=1

22 J. A. BALL AND V. BOLOTNlKOV

Since the right-hand side expressions in (3.42) and (3.43) are identical, 3.40 f0llows. Now we have from (3.40)

(I, (1 - A* Zdlag(Z)*)-lC*y) = (C(1 - Zdiag(Z) A)-1 f, y) = «sf) z ,y

= /,T·,zy

for every z E lDid and f E EBt=l 'H.(Ki ), and thus, equality (3.31) holds. 0

Step 2. Given operators A, C and D with A 0/ the form 3.17) equal to contractive solution of the structured Gleason problem for the kernel collechon {Kl, ... ,Kd} and with C and D given by (3.21), ifU = (~Zl t.s a ~e

realization of S for some operator B = EB B,: U ~ ffi~=1 'H.(K.), then B salta jh,

EB~=l1i(Kk)-Gleason problem for S, i.e., B satisfies identity 3.18.

PROOF OF STEP 2. Since U = [~Z] is a realization for S, equalrty 3.33 holds. Making use of equality (3.31) (which holds by Step lone can write 3.33 as

B* Zdiag«)*'lI.'(·, ()y + D*y = S ( .y

or, in view of formula (3.21) for D, as

(3.44) B* Zdiag«r'lI.'(·, ()y = S«)*y - S 0 .y.

Taking the inner product of both parts in (3.44) with an arbitrary funw n J E EBt=l1i(Ki) leads us to Zdiag(z)Bu = S(z)u-S(O u which is the same as 3.18 0

To complete the proof of the theorem, it suffices to show that there exists an operator B : U -+ EBt=l 1i(Ki) such that equality (3.33 holds for every u E U and the operator matrix

(3.45) U* = [A* C*1. r ffit=1 'H.(K·)1 ~ r ffi~=1 'H. K )1 B* D* . l y l u

is a contraction. As we have seen, equality (3.33) is equivalent to (3.44 , v-htch in turn, defines B* on the space 'D introduced in (3.13). Let us define B: 1> -i

EB:=l1i(Ki ) by the formula

B: Z«)*'lI.'(.,()y = S«)*y - S(O)·y

and subsequent extension by linearity and continuity; it is a consequence of the isometric property of the operator V in (3.24) that the extension is well-defined and bounded. We arrive at the following contractive matrix-completion problero:

d -find B: U -+ EBi=l 1l(K,) such that B*!'D = B and such that U· of the form (3.45 is a contraction. Following [10] we convert this problem to a standard matrixcompletion problem as follows. Define operators

d d

Tn: Vi -+ E91i(K,), T12: 'D ~y ~ E9ll(K,), .-1 \=1

by

(3.46)


Identifying we then can represent U· from (3.45) as

(3.47) 1>.1. -+ EB~=l 1i(K,) 1>Ef)Y U

where X = B*\v.L is unknown. Thus, an operator B gives rise to a canonical functional-model realization U = t ~ g 1 of S if and only if it is of the form

X. 'D.1. It

B = B* : U -+ 'D ~ E!11i(K,) ,=1

where X is any solution of the contractive matrix-completion problem (3.47). But this is a standard matrix-completion problem which can be handled by the wellknown Parrott's result [291: it has a solution X if and only if the obvious necessary conditions hold:

3.48

:Making use of the definitions of Tn. T12 , T22 from (3.46), we get more explicitly

Tt2 A"\v C" [Tu T121 = [A* Co01. T22 = B D'"

Thus the first in (3.48) is contractive since A is a contractive solution of the stru tured Gleason problem (3.15), while the second expression collapses to -V see f rmula 3.24) which is isometric by (3.1). We conclude that the necessary

nditi ns 3.48 are satisfied and hence, by the result of [29], there exists a solution X to problem 3.47. This completes the proof of the theorem. 0

4. de Branges - Rovnyak kernels associated with a Schur - Agler-class function on a domain with matrix-polynomial defining function

A generalized Schur class containing all those discussed in the previous sections as special was introduced and studied in [4,9] (see also [5] for the scalar-valued

and can be defined as follows. Let Q be a p x q matrix-valued polynomial

4.1

such that

4.2

Q(z) = qu (z) .. . q1q(Z)

• , ,

• • •

Q(O) = 0

and let 'DQ E en be the domain defined by

'DQ = {z E en: \\Q(z)1I < I}.

Now we recall the Schur Agler class SAQ (U, Y) that consists, by definition, of £. U, Y)-valued functions S(z) = S(Zl" '" zn) analytic on 'DQ and such that IIS(T) II ::; 1 for any collection of n commuting operators T = (T1, ... , Tn) on a. Hilbert space K:, subject to Q(T)U < 1. By [5, Lemma 1], the Taylor joint spectrum of the commuting n-tuple T = (T1, .•. , Tn) is contained in 'DQ whenever IIQ(T)II < 1, and hence SeT) is well defined by the Taylor functional calculus (see [19]) for any £. U, Y)-valued function S which is analytic on 'DQ. Upon using K: = e and T, = z,

J. A. BALL AND V BOLOTNIKOV

for j 1, ... , n where (Zh"" zn) is a point in VQ we conclude that any fun<'t SAQ(U,Y) is contractive-valued, and thus, the class SAQ U,Y) is th Ia the Schur class SPQ (U, Y) of contractive valued functions analyt' c on 'DQ B von Neumann result, in the case when Q(z) = z, these classes coinClde- in getll!r:!l. SAQ(U, Y) is a proper subclass of SPQ(U, Y). The following result appears (see also [5] for the scalar-valued case U - Y = C) and is yet another m b\

analogue of Theorem 1.1. We will often abuse notation and will wnte Q z lIlStead of Q(z) ® 1 where 1 is the identity operator on an appropnate Hilbert space from the context. When the following theorem is viewed as a parallel fTh rem we see that, just as in the polydisk setting, there is no parallel to rut lIS

and (lb).

Theorem 4.1. Let S be a £(U, Y)-valued fund on defined on'DQ The lowing statements are equivalent:

(4.3)

(1) (c) S belongs to SAQ(U,y), (2) There exists a positive kernel

(1K~1 .. . lK~p1 IK =: :: 'DQ

1Kpl IKpp

which provides a Q-Agler decomposition f r S .e, su that f z,( E'DQ,

P q

(4.4) Iy - S(z)S«()* = L IKkk (Z, () - L L q z QT~ z ( k=1 k-l,=1

(2') There exist an auxiliary Hilbert space X a d a fu t&

(4.5) H(z) = [Hl(Z) '" Hp(z]

analytic on'DQ with values In .c(XP,)7) so that f revery z, ( E 'DQ

(4.6) Iy - S(z)S«()* = H(z)(Ix - Q z Q ( * H ( *.

(4.7)

(3) There exist an auxiliary Hilbert space X and a u ~ tary ronnec g perot U of the form

u = [~ ~]: [~] ~ [~ ] so that S(z) can be realized in the form

(4.8) S(z) - D + C(Ixp - Q(z)A) lQ(z)B for all Z E'DQ.

(4) There exist an auxiliary Hilbert space X and a contractive connecun9 r erator U of the form (4.7) so that S(z) can be realized in the form (4.

Remark 4.2. If S = [~!~ ~~~] E SAQ(U1 $U2,)71 $Y2), then the block~t~ 5'J belongs to the Schur Agler class SAQ(U"YJ ) for i,j = 1,2. For the pro '

suffices to note that IIS'J(T)II $ IIS(T)\\.


Remark 4.3. The equivalence (2) .: :. (2') can be seen by using the Kolmogorov decomposition for the positive kernel oc:

(4.9) OC(z, () = • • •

Hp(z)

The implication (4) > (1) can be handled by any of the three approaches sketched in Remark 2.2. Following the approach from (10), we first handle the case where U is coisometric, using the identity

(4.10) 1- S(z)S«()* = C(I - Q(z)A) 1(1 - Q(z)Q«()*) (I - A·Q«()*)-lC·

•

holding for S of the form (4.8) and U given by (4.7), the straightforward verification of which is based on the identity

[CeI - Q(Z)A)-lQ(Z) I) U = (C(I - Q(Z)A)-l S(z)).

Then the general (contractive) case follows by extension arguments and Remark 4.2.

Remark 4.4. With no assumptions on the polynomial matrix Q(z) some degener ies ccur which ca.n be eliminated with proper normalizations. We note first fall th t it is natural to assume that no row of Q(z) vanishes identically; other--

wise one can cross out any vanishing column to get a new matrix polynomial Q(z) f smaller size which defines the same domain VQ in en. Secondly, in the second

term of the Q-Agler decomposition (4.4), the (i, l)-entry lK.;,1 of OC is irrelevant for any pair of indices i,l such that at least one of qik(Z) and qlk(Z) vanish identically f r each k = 1, ... , q. Note that if the first reduction has been carried out, then all wag nal entries oc.. are relevant in the second term of (4.4) in this sense. It f llows that, without loss of generality, we may assume that OCil(z,() == 0 for each such pair of indices (i, l). To organize the bookkeeping, we may multiply Q(z) on the left and right by a permutation matrices II and II' (of respective sizes P x P and q x q so that Q z) = IIQ(z)II' has a block diagonal form

Q(l)(z) 0 -

4.11 Q(z) = • • •

o Q(d)(z)

with the ath block Q a) (0: = 1, ... ,d) of say size POI x qOl and of the form

Q (a) (z) = [q(OI) (z))P.:: ~~ ',3 .-13-1

-and irreducible in the sense that Q has no finer block-diagonal decomposition after perll!utation equivalence, i.e., for each 0: for which Q(a) is nonzero and for any pair of indices i, l (1 :5 i, I :5 POI)' there is some k (1 :5 k :5 qj) so that either q.~ (z) or 'lJ~)(z) does not vanish identically. Without loss of generality we may -assume that the original matrix polynOInial Q is normalized so that Q == Q. We may then aSS11me that the positive kernel in (4.3) and (4.4) has the block diagonal

~6 J. A. BALL AND V. BOLOTNIKOV

decomposition

(4.12) llK(I) (Z, ()

lK(z,() = o

where lK(a) in turn has the form

llK(a)

11

lK(a) = : lK(a)

p",l

Under the normalizing assumption that lK has this block diagonal f rm -1.12 Q(z) is written as a direct sum of irreducible pieces (4.11 ,the nstruct1 us to follow can be done with more efficient labeling but at the cost of an additI nal \1lf

of notation. We therefore shall assume in the sequel that this diag nal structure • not been taken into account (or that the matrix polynomial Q is already irredu e until the very end of the paper where we explain how the polydisk settIDa can seen as an instance of the general setting.

As in the previous particular settings of the ball and of the polydIsk, we mtro-duce the weak-coisometry property as the property equivalent to 4.10 llapsmo to

1 - S(z)S«)* = e(1 - Q(z)A)-I(I - Q(z)Q (* 1-A*Q ( • -Ie-. Definition 4.5. The operator-block matrix U of the form 4.7 is weakl lSO-

metric if the restriction of U" to the subspace

(4.13) .- V lQ«)"(1 - A*Q«)* -lc·y1 (Xq1 'Du··- c y (EVQ Y yEY

is isometric.

Due to assumption (4.2), the space 'Du. splits in the form 'Du. = 'D EB Y where

(4.14) 'D = V Q«)*(I - A*Q«)*)-IC"y c X q•

(EVQ,yEY

4.1. Weakly coisometric canonical functional-model Q-realizations. Let us suppose that we are given a function S in the Schur Agler class SAQ u,Y together with an Agler decomposition lK. as in (4.3) (so (4.4) is satisfied). We'9l,u use the notation Q.k«) for the k-th column of the polynomial matrix Q. What actually comes up often is the transpose:

(4.15) Q.k«)T = (qlk«) q2k«) ... ~k«)]' Note that with this notation the Q-Agler decomposition for S (4.4) can be written more compactly as

p q

(4.16) 1:v - S(z)S«)* = L KIt,k(Z, () - L QJ)(z)lK(z, ()QJJ«)" It I j 1

an expression more suggestive of the Agler decomposition (3.1) for the pol)~ case.


We say that the operator A: ll(K)P ~ 1l(K)9 solves the Q-coupled Gleollon problem for 1-£ (K) if

P q

(4.17) Z)!k,k(Z) - !k,k(O)) = L Q.k(z)T[A!1k(Z) for all I E ll(K)P 10=1 10=1

so each f E ll(K)P has the form

1= It

• • •

where 110 = • • •

!k,P E ll(K).

Similarly, we say that the operator B: U ~ ll(K)q solves the Q-coupled ll(K)Gleason problem for S if the identity

9

(4.18 S(z)u - S(O)u = L Q.k(z)T[Bu)k(Z) holds for all u E U. 10=1

The following proposition gives the reformulation of Gleason-problem solutions in terms of the adjoint operators. In what follows, we let {ell .. " ep } to be the standard basis for CP.

Proposition 4.6. The operator A: ll(K)P -+ ll(K)q solves the Q-coupled Gleason problem 4.17) if and only if the adjoint A* of A has the following acnon on specrol kernel functwns:

4.19 • • •

• • •

-

fo all (E VQ and y E y, where Ei = Iy ® e, for i = 1, ... ,p:

Iy 0 o Iy

4.20

• • •

o • • •

o Iy

•

The opemtor B: U -+ 1l(K)9 solves the Q-coupled ll(K) -Gleason problem (4.18) for S 'if and only if B*: ll(K)q -+ U has the following action on special kernel functwns:

(4.21 B*: • • •

H S«)"'y - S(O)*y for all C E VQ and y E y. K(., ()Q.q«)T*y

PROOF. We start with the identity

K(z,()E1y

(4.22) Q«)"' • • •

-- • • •

which holds for all z, ( E V Q and y E Yj once Q«)* is interpreted as Q«)*®Iy and similarly for Q.,.«)T*, this can be seen as a direct consequence of the definitions


(4.1), (4.3), (4.15) and (4.20). Letting

(4.23) 1l'( z, () := : llK(Z' ~)E1J

lK(z, ()Ep

for short, we then can write formulas (4.19), (4.21) more compactly as

(4.24) A*Q«)*1l'(·, ()y = 1l'(', ()y -1l'(" O)y,

(4.25) B*Q«)"1l'(·, ()y = 8«)*y - 8(0)*y

where now Q«)* is to be interpreted as Q«)" ® 11£(lK)' In the following computations, Q«)* is either Q«)" ® Iy or Q«)" ® 11£(K) according to the context. B)

the reproducing kernel property, we have for every f = EI):=1 I" E 1llK. P, p p

(4.26) (I,'lr(',()Y)l£(IK)P = LU",lK(., () E"y) l£(K) = I: Ei.f,,« ,y y "=1 "=1

Therefore,

(4.27) (I,1l'(·,()y-1l'(·,O)Y)l£(IK)P = <t;u",,,«)-f,,,,, 0 ),Y)y'

On the other hand, it follows again from (4.26) that

(I,A*Q(z)*1l'("Z)Y)l£(IK)P = (Q(z)AI,1l'("Z)Y)l£(K)P = <t,[Q(Z)AllJJ Z ,Y y

and since p p q

(4.28) ~[Q(z)Alkj(z) = L L <l.i"(z)[AJ],,,](z) j=1 j=I,,=1

we get

(I, A*Q(z)"1l'(·, z)y) 1£ (lK)p = < t Q.,,(z) T[Afl,,(z), Y) y' "=1 Y the

Since the last equality and (4.27) hold for every f E 1l(K)P, ( E VQ and y ~val~ce equivalence of (4.17) and (4.24) (which is the same as (4.19)) folloWS. EqUl

of (4.18) and (4.25) follows by the same argument from equalities

(1£, S«)"y - S(O)"y)u = (8«)1£ - 8(0)1£, y}y

and (1£, B*Q(z)*1l'(·, z)y)u = (Q(z)Bu, 1l'(', Z)Y)l£(IK)P

= (t [Q(z) Bul j ,j (z),y) = (:tQ.,(z)T[Bti1.(M), 3=1 y "==1 Cl

holding for all 1£ E U and y E :V.


Just as in the particular cases discussed in the previous sections, it turns out that the formulas (4.19) and (4.21) can be extended by linearity and continuity to define uniquely determined bounded well-defined operators

B;: V-+U

as a consequence of the isometric property of the operator V defined below in (4.33).

Definition 4.7. We say that the operator-block matrix U = l ~ ~) : 1i(OC)P EB U -+ 1i(JK)'l EB Y is a canonical functional-model colligation matrix for the given function S and Agler decomposition JK if

4.29

1) U is contractive. 2) The operator A solves the Q-coupled Gleason problem (4.17) for 1i(OC). 3) The operator B solves the Q-coupled 1i(OC)-Gleason problem (4.18) for S. 4 The operators C: 1i(OC)P -+ Y and D: U -+ Yare given by

C: l1(z)

• • •

t-+ il,l(O) + ... + fp,p(O), D: u t-+ S(O)u.

F flnulas 4.25) can be written equivalently in terms of adjoint operators as follows:

4.3 C*: y t-+ 1'(" O)y D* : y t-+ S(O)*y

where T is defined. in (4.23). The next theorem is the analogue of Theorem 3.7.

Theorem 4.8. Let S be a given function in the Schur - Agler class SAQ (U, Y) and suppose that we are given an Agler decomposition (4.4) for S. Then there exists a canonical functwnal-model colligation associated with the kernellK..

PROOF. Let us rearrange the given Agler decomposition (4.4) or (4.16) as

q p

Iy + L Q.k(z)TOC(z, ()Q.k«) T* = S(z)S«)* + L EjOC(z, ()Ej, 10=1 j=l

and then invoke the reproducing kernel property to rewrite the latter identity in the inner product form as

q

4.31) ~ (lK.(., ()Q.k«) T .. y , lK.(', Z)Q.k(Z) T*y'ht(K) + (y, y'}y 10=1 p

= ~ (lK.(., ()EjY, OC(" z)E,Y')1{(K) + (S«)*y, S(z)*y')u. 3-1


The latter can be written in the matrix form as

(4.32) / rQ«()*1l'(·, ()Y1 ' rQ(Z)*1l',(.,Z)Y'1) \ L Y L Y 'H(K)geY

( fT("( y1 fT ',Z y'1 = l s (ty , l S Z *y'

The latter identity implies that the formula

(4.33) V: rQ«)*1l'("()Y1 r-+ rT "()y1 L y l S ( .y

extends by continuity to define the isometry from 1\ = V ffi Y c 'Hex q Ee Y see (4.14) for definition of 'D) onto

nv = V l~~(f:~ 1 c l1l~)p1· (E'DQ lIEY

Let us extend V to a contraction U*: ['Hc;)9 1 ~ ['H~)P]. Thus,

(4.34) U* - [A* c*1· fQ«)*T(., ()y1 r-+ fT("()Y1 - B* D* . l y lS«)*y •

Computation of the top and bottom components in (3.25) gives

(4.35) A*Q«)*1l'(., ()y + C*y = T(·, ()y,

(4.36) B*Q«)*T(·, ()y + D*y = S«)*y.

Letting ( = 0 in the latter equalities and taking into account (4.2) leads ~ tOt~4:Oof from which we see that C and D are of the requisite form (4.29). Substltu

10 uiv(4.30) into (4.35) and (4.36) then leads us to (4.24) and (4.25) w~ch a:n:ucal alent to (4.17) and (4.18), respectively. Thus we conclude that U 18 a 0 functional-model colligation as wanted.

. an operatol' For this general setting we define observability as follows: gI~n called Q.

A: XI' -+ xq and an operator C: XI' ~ Y, the pair (C, A) will. beb

hOod of observable if the identities C(I - Q(Z)A)-II.;x = 0 for all z in a neigh or denote the origin and for all i = 1, ... ,p forces x = 0 in X. By Z. : X -+ X" we

TRANSFER-FUNCTION REALIZATION a1

the inclusion ma.p which embeds X into the i-th component of XI' = X ffi .•. ~ X. Thus

0 Xl

• • • • • •

(4.37) ~: x,t-+ X, and r· • • X, t-+ X •.

• • • • • •

0 xI'

The Q-observability can be equivalently defined in terms of adjoint operators as

(4.38 VrI;(I A*Q(z)*) lC*y:zEb.,yEY,i=1, ... ,p}=X

where b. is some neighborhood of the origin in en. The following theorem is the analogue of Theorem 1.2 for the present general setting.

Theorem 4.9. Let S be a function in the Schur-Agler class SAQ(U, Y), let the poS\uve kernellK. of the foun (4.3) provide an Agler decomposition (4.4) for S a d suppose that U = t~ Z) : ll(lK.)p ffi U -t ll(lK.)q ffi Y is a canonical functionalmodel lltgatton assoctated with 8 and lK.. Then the following hold:

1 U 1.8 eakly C01.8ometnc. 2 The p (C,A) 1.8 Q-observable in the sense of (4.38).

439

3 "' reco er S as 8(z) = D + C(I - Q(z)A)-lQ(z)B. 4 If U = ~ ~ : XP ffi U -t xq ffi Y is another colligation matrix enjoying

p ert es (1), (2), (3) above, then there is a canonical functional-model -col 'g tum U for (8, lK.) such that U and U are unitarily equivalent in the sense that there is a unitary operator U: X -t 1l(lK.) so that

A B C D

EI1~=l U o

o _ EI1?=l U 0 Iy - 0 Iu

- -A B CD'

PROOF. Let U = t~ Z) be a. canonical functional-model realization of 8 associated with a fixed Agler decomposition (4.4). Then combining equalities (4.24), 425 equivalent to the given (4.17) and (4.18) by Proposition 4.6) and also for

mulas 4.30 equivalent to the given (4.29)) gives

4.40 1'("( y = (I - A*Q(()*)-l']['(., O)y = (I - A*Q(()*)-lC*y

and

4.41 S(()*y = 8(0)*y + B*Q(()*']['(', ()y = D*y + B*Q(()*,][,(., ()y.

Substituting (4.40) into (4.41) and taking into account that y E Y is arbitrary, we get

4.42) 8(()* = S(O)* + B*Q(()*(I - A*Q(()*)-lC·

which proves part (3) of the theorem. Also we have from (4.40)

V r.(I - A*Q(()*)-lC·y = V :r;']['(.,()y C;E"DQ IIEY.

t-l, .... ,p CE"DQ,IIEY,

t 1, .. 'IP

32 J. A. BALL AND V BOLOTNIKOV

and we can proceed due to (4.37) and (4.23) as follows:

V X;1l'(.,()y= V JK(·,()E.V= V K(',()y=1£X C;EDQ,lIEY,

i-I, ... ,,, C;E'DQ,lIE Y,

1.=1, ... ,1' <E'DQ yEY

Thus the pair (C, A) is Q-observable in the sense of (4.38). On the other ban equalities (4.24), (4.25) are equivalent to (4.34). Substituting (4.40 into 4.34 an into identity (4.32) (for z = ( and V = V') gives

• [Q()'(I - A*Q()*)-IC*v1_ r(1- A*Q ( • -1C*V} u y - l S < 'y

and

respectively. The two latter equalities show that U· is isometric on the space Vu (see (4.13)) and therefore U is weakly coisometric.

To prove part (4), let us assume that

(4.43) 8(z) = 8(0) + O(I - Q(z)A)-IQ(z B is a weakly coisometric realization of 8 with the state space X and such that the pair (0, A) is Q-observable in the sense of (4.38). Then

1- S(z)S()* = O(I - Q(z)A)-l(I - Q(z)Q (t 1- A*Q ( * -Ie

which means that 8 admits a representation (4.6) with H z = 0 1- Q z A -Let I; be given as in (4.37). Representing H in the form 4.5 with

(4.44)

we then conclude from Remark 4.3 that 8 admits the Agler decompositi n 44 with

for i,j = 1, ... ,p.

Let 1£(IK) be the reproducing kernel Hilbert space associated with the posithe kernellK = [1K'31~,j=I' Let us arrange the functions (4.44) as follows

(4.45) (

HI.(Z)] lo(I -Q(.Z)A)_lXl] G(z) := : = : .

Hp(z) O(I - Q(z)A)-IIp

Since by construction lK(z,() = G(z)G()* and since (0,...1) is Q-observable, the formula

(4.46) U: x ~ G(z)x

defines a unitary map from X onto 1£(IK). Let us define the operators A: ll(iW-+ 1£(IK)q and B: U -+ l£(lK)q by

(4.47) and B= (a,U)B. , 1


In more detail, using representations

- - -All ••• All' Bl

- -A= • - : XI' ~ xq and B= • • • • • • • - - -Aql ••• Aqp Bq

we define

Au • •• All' and B= A= • •

• • • •

Aql ••• Aqp

block-entrywise by

4.48) and

• • •

: U -i- xq,

33

for i = 1, •.. , q and j = 1, ... ,p. We next show that the operators A and B sol~e the Gleason problems (4.17) and (4.18), respectively. To this end, take the genenc element J of 1£ (K)P in the form

4.49 J(z) = • • •

where x:= • • •

On account of 4.45), we have for f and x as in (4.49),

P I'

L: lk.k(Z) = L: O(l - Q(z)A)-lIkXk k=l k=1 P

= C(I - Q(z)A)-lL:IkXk = O(l - Q(Z)A)-lx. k=l

Therefore, aud since Q(O) = 0, we have

p

4.50 L:(Ik.k(Z) - A,k(O)) = O(I - Q(z)A)-lx - Ox k=1

On the other hand, we have by (4.45) and (4.48),

P

{AflkJ(Z) = L:Ak,G(z)x, .=1 j

p p

= G(z) LAkiX, == O(l - Q(Z)A)-lIj L:AkiXi ,==1 j i=1

and it follows directly from (4.37) and (4.1) that

P q P

L z:: 'l.7k(z)I, L AkiXt == Q(z)Ax. 3=1 k=l s=1


Making use of the two last equalities and of (4.28) we get q P q

L Q.k(Z) T[Aflk(z) = L: L: <l.ik(z)[AJ1k,j(Z) k=l j=lk=l

P q P

= L: L: <l.ik(z)0(1 - Q(Z)A)-lI, L:Ak,x. j=l k=l ,=1

P q P

= O(I - Q(Z)A)-1 LL<l.7k(Z)I, L:Ak,x, j=l k==l 0=1

== 0(1 - Q(z)A)-lQ(z)Ax

which together with (4.50) implies (4.17). Similarly we conclude from (4.15), (4.45 • (4.47) and (4.43) that

q P q

LQ.k(Z)T[Bu]k(Z) = LL<l.ik(z)[Bku],(z) k=l j=lk=l

P q

= L L <l.ik (z) [G(Z)BkU]j j=lk=1

P q

= L L <l.ik(Z)O(I - Q(Z)A)-lI,Bku j=lk=l

P q

= 0(1 - Q(Z)A)-l L L <l.7k(z)I,Bku j=lk=l

= 0(1 - Q(z)A)-lQ(z)Bu = S(z)u - S(O)u

and thus, B solves the Gleason problem (4.18) for S. Finally, for f and:c of the form (4.49), for the operator U defined in (4.46) and for the operator C defined on 1l(lK)P by formula (4.29), we have

P P P P

C(EBU)x = Cf = Lik,k(O) = L O(I - Q(O)A)-lIkXk = LOIkXk = ex, i=l k=l k=l k=l

and thus,

C(~U) =0.

The latter equality together with (4.47) implies (4.39). According to Definition 4.7, the colligation U = [~ g] is a canonical functional-model colligation associated with the Agler decomposition lK of S. 0

Let us say that the operator A: 1l(lK)P --+ 1l(IK)q is a contractive solution of the Q-coupled Gleason problem for 1l(1K) if in addition to identity (4.17) the inequality

k

I\Afll~(K)4 ~ IIfl\~(K)P - Ll\fk,k(O)II~ , 1

holds for every function f e 1l(lK)P or equivalently, if the pair (C, A) is contractive where C: 1l(lK)P ~ Y is the operator given in (4.29).


Theorem 4.10. Let (4.4) be tl fixed Agler decomposition of tl given function S E SAQ (U, Y) and let 0 and D be defined as in (4.29). Then

(1) For every contractive solution A of the Q-coupled Gleason problem (4.17), there is an operator B: U ~ 1i(lK)q such that U = l ~ ~ 1 is a canonical functional-model colligation for S.

(2) Every such B solves the Gleason problem (4.18) for S.

The proof is very much similar to the proof of Theorem 3.10 and will be omitted. In conclusion, we compare functional model Q-realizations obtained in this

section with particular cases considered in Sections 2 and 3.

The unit ball setting. In this case, Q = Zrow (in particular, p = 1) and definition 2.2) can be interpreted as the (uniquely determined) Agler decomposition of the form (4.4) with the kernellK= Ks. Then (4.23) gives 'll'(z,() = Ks(z,() and 4.33 coincides with (2.15). Since all canonical functional-model colligations are

obt.ained via contractive extensions of isometries V (from (2.15) for the unit ball setting or from (4.33) for the general Q-setting), it follows that realizations constructed in Section 2 ca,D be obtained from those in Section 4 by letting Q = ZIOW'

Moreover, if Q = Zrow, then observability in the sense of (4.38) collapses to observability defined in part (2) of Theorem 2.9.

The unit polydisk setting. In this case, Q = Zdiag, P = q = d, and the Agler representation (3.1) for an S E SAd(U,y) can be written in the form (4.4)

Kl 0

with the kernel ][{ = • • •

4.51

. Then (4.23) takes the form

'll'(Z, () = • • •

Kd(Z, () I8l ed

where {eh--' ,ed} is the standard basis for ICd. Observe that (4.51) is not the same as 3.10 . Now 4.33) collapses to

(lKl(Z, () I8l el Kl(Z, () I8l el

4.52) V: • • •

(dKd(Z, () I8l ed y

whereas 3.24) can be written as

(lK1(z,() • -4.53 V: , •

(dKd(Z, () y

, • •

Kd(Z, () I8l ed S«()"y

K1(z, () • , ,

Kd(Z, () S«()*y

-To get canonical functional-model realizations as in Definition 3.5, we extend V to . - A" C" d ( d a contractIOn U" = B" D" : EB.=l1i Ki ) EB Y ~ EB i=l1i(Ki) EB u. If for such a

contraction we let U to be of the form (3.2) with

- -A., = A., I8l e.e;, OJ = OJ I8l ej,


then U· will be a contraction from E9~=1 (l£(K.))P ffi Y to EB~=1 (1£ (K.))P 9 U extending the isometry V given in (4.53). It is not hard to see that U is a canonical functional-model Q-realization for S in the sense of Definition 4.7. Thus, any "polydisk" canonical functional-model realization gives rise to a canonical functionalmodel Q-realization for S. Of course, the converse is not true.

To see the polydisk setting as a particular instance of the general Q-setting we need to make use of the block-diagonal decomposition of Q into irreducib e parts discussed in Remark 4.4. For the polydisk setting with Q z) = ZdIag Z ,

this diagonal structure is nontrivial and already apparent. Thus we assume that Q(z) has the form (4.11) and the positive kernel OC giving rise to the Q-Agler decomposition (4.4) has the compatible block decomposition (4.12 . The Q-Agler decomposition (4.4) now has the form

(4.54) I - S(,)S«), ~ t.( E JKl-:> (', () - t 1:, i: (,)q,: ()K,;' z,( 1 and can be rewritten in inner-product form as

d qc>

L L (oc(a) (', ()Q~~) «() T.y , oc(a) (', Z)Q.k(Z) T" y ')ll(K

a=1k=1 d Pc>

= L L(lK.(a) (', ()E3a)y, oc(a) (', z)Eja)Y'}ll(K" ) + s (ry, S Z 'Y1U

a=1j=1

where E3a) = Iy ® ej and where {el, ... ,ep,J is the standard basis for 0' ... Then

the isometry V in (4.33) has the form

(

K(Q) (%.{)EiQ

) 1 where T(a) (z, () = : and where V has domain equal to Dv :=: 'D al Y

K(a) (%.{)E~:> where

Rv = V l1t(:~'()·1 c 41£(oc(a)PQ EBUi cevQ.lleY T( )(',C)y 0-1

S«()*y

all this specializes to (4.53) for the polydisk case. We say that the operator A: EB~ 11£(lK(a)PQ -+ E9: 11£(lK(a)qQ solves the Q-structuTed Gleason problem

TRANSFER.FUNCTION REALIZATION

for the kernel collection {K(I), •.• ,K(d)} if

for all f =

d Pa d qa

L L(J~~2(z) - f~~2(0)) = 1: L Q~~)(z)TlAf1~a)(z) a=lk=1 a-lk 1

1(1)

• • • • • •

'

(a) I(a) p" k,p",

37

We say that the operator B: U -+ E9!=lll(K(a»)q", solves the Q-structured {K 1), .... K d)}-Gleason problem lor S if

d qa S(z)u - S(O)u = L L Q~~) (z)[Bu1~a) (z)

a=1 k=l

for all u E U. where we write Bu =

[Bult

• • •

[Bu](a) k,l

{Bu1 a = : where in turn (Bul~a) = : E ll(K(a»). We de-• •

[Bu] " [Bu](a) 0: k,po

fine a canonical functional-model colligation matrix U for a given function S E SAq U.Y and left Q-Agler decomposition {K1, ... ,Kd} (so (4.54) holds) to be any perator-matrix U = {~gl: E9!=lll(K(a»)Pa ffiU -+ E9!=lll(K(a»)qa ffiY so that

1 U is a contraction. 2 the operator A solves the Q-structured Gleason problem for the kernel

collection {K(I) •...• K(d)}. 3 the operator B solves the Q-structured {K(1), ... , K(d)}-Gleason problem

for S, and 4 the operators C and D are given by

11(a)

d d Pa

C:ffi : ~LLlt2(0), D:u~S(O)u. a=1

'

(a) Pc>

a=lk=1

Then we leave it to the industrious reader to check that Theorems 4.8, 4.9 and 4.10 all go through with this block-diagonal modification. Specializing this formalism to the polydisk c.ase picks up exactly the results of Section 3.

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30. W. Rudin, FUncuon theory m the umt ball oj en I Grundlehren Math. Wiss., vol. 241, Springer, New York, 1980.

31. D. Barason, Sub-Hardy Hilbert spaces m the unit dISk, Univ. Arkansas Lecture Notes Math. Sci., vol. 10, Wiley, New York, 1994.

DEPARl'MENT OF MATHEMATICS, VIRGINIA TECH, BLACKSBURG, VA 24061-0123, USA

E-mail address: beUlDmath. vt. edu

DEPARl'MENT OF MATHEMATICS, THE COLLEGE OF WILLIAM AND MARY, WILLIAMSBURG VA

23187-8795, USA E-mail address: vladilDmath.vm.edu

Centre de Rl'Cherchelo M&thematlque. CRM Proceedings and J Bcture Notes Volume 51, 2010

Two Variations on the Drury-Arveson Space

Nicola Arcozzi, Richard Rochberg, and Eric Sawyer

1. Introduction

The Drury - Arveson space DA is a Hilbert space of holomorphic functions on Bn +!, the unit ball of Cn +!. It was introduced by Drury [11] in 1978 in connection with the multivariable von Neumann inequality. Interest in the space grew after an influential article by Arveson [7], and expanded further when Agler and McCarthy [1] proved that DA is universal among the reproducing kernel Hilbert spaces having the complete Ne a- Pick property. The multiplier algebra of DA plays an important role in these studies. Recently the authors obtained explicit and rather sharp estimates for the norms of function acting as multipliers of DA [3], an alternative proof is given in [17].

In our work we made use of a discretized version of the reproducing kernel for DA, or, rather, of its real part. In this note we consider analogs of the DA space for the Siegel domajn, the unbounded generalized half-plane biholomorphically equivalent to the ball. We also consider a discrete model of the of the Siegel domain which carries a both a tree and a quotient tree structure. As sometimes happens with passage from function theory on the disk to function theory on a halfplane, the transition to the Siegal domain makes some of the relevant group actions more transparent. In particular this quotient structure, which has no analog on the unit disk i.e., n = 0), has a cleaner presentation in the (discretized) Siegel domain than in the ball.

Along the way, we pose some questions, whose answers might shed more light on the interaction between these new spaces, operator theory and sub-Riemannian geometry.

We start by recalling some basic facts about the space DA. An excellent source of information is the book [2]. The space DA is a reproducing kernel Hilbert space

2000 Mathematu:s SubJect Cla.s8Jficat,on. Primary 46E22j Secondary 32A37. The first author partially supported by the COFIN project Analisi Armonica, funded by the

Italian Minister for Research. The second author's work work supported by the National Science Foundation under Grant

No. 0070642. The third author's work was supported by the National Science and Engineering Council of

Canada

@2010 American Matbematlcel SOciety

41

42 N. ARCOZZI ET AL.

with kernel, for z, W E 'lBn+1

(1.1) 1

K(z,w) = -1--w·z

Elements in DA can be isometrically identified with functions / holomorphic in Bn+1, J(z) = LmE",,,+l a(m)zm (multiindex notation), such that

2 '" m! 2 IIJIIDA = L..J Iml!la(m)I < 00. mEN,,+l

When n = 0, DA = H2. the classical Hardy space. The multiplier algebra f H2, the algebra of functions which multiply H2 boundedly into itself, 18 Hoo, the algebra of bounded analytic functions. In general the multiplier algebra Af(DA f DA is the space of functions 9 holomorphic in 18n +l for which the mult plicatlon operator /H g/ from DA to itself has finite operator norm which we den te by IlgIIM(DA)' For n > 0, M(DA) is a proper subalgebra f H oo

, however in BOme ways it plays a role analogous to Hoo. In particular the multiplier norm 9 M(DA

replaces the HOO norm in the multivariable version of von Neumann's InequalIty [ll}. Also, the general theory of Hilbert spaces with the NevanlinDa- Pick. property exposes the fact that many operator theoretic results about H2 and H OO are speaal cases of general results about Hilbert spaces with the Nevanljnna- Pick property, for instance DA, and the associated multiplier algebra.

Given {wi If'=l in Bn+1 and {A, If'=l in C, the interpolation problem f findm" gin M(DA) such that. g(Wj) = Aj and IIgI\M(DA :s I, has soluti n if and only If the "Pick matrix" is positive semidefinite,

[(1 - wjwh)K(A" Ah)l~\-l ~ O.

Agler and McCarthy [1] showed that the (possibly infinite dunensional DA kernel is universal among the kernels having the complete Nevanlinna Pick property, ~hich is a vector valued analog of the property just mentioned. While for n = 0 we have the simple characterization I/gl/M(DA) = gl M H2) = 9 Hoo, no such formula exists in the multidimensional case. However, 8 sharp, geometric estimate of the multiplier norm was given in [3].

Theorem 1. (A) A function g, analytic In 18n +1, is a mulhpller Jor DA if and only iJ 9 E HOC and the measure tJ = tJg, dtJg := (1- z 2) Rg 2 dA(z) as " Carleson measure Jor DA,

(1.2)

Here dA is the Lebesgue measure in 'lBn+1 and R is the radial differentiation operator. In this case, with K(tJ) denoting the infimum 0/ the possible C(tJ) in the previous inequality,

IIgIlM(DA) s::; IIgl\Hoo + K(tJ)1/2

(B) For a in 'lBn+1, let S(a) = {w E lBn+1 : 11 - a/lal' wi :s (1-laI2)} be tht Carles on box with vertex a.

Given a positive measure tJ on 18n +1. the follOwing are equivalent: (a) p. is a Carles on measure Jor DAj

TWO VARIATIONS ON THE DRURY ARVESON SPACE

(b) the inequality

Re K(z, w)cp(z)cp(w) d/-l(z) d/-l(w) ~ C(/-I) B,,+l

holds for all nonnegative cpo (c) The /-I satisfies both the simple condition

(SC) /-I(S(a» ~ C(/-I)(1-laI2)

and the split-tr-ee condition, which is obtained by testing (2) over the characteristic functions of the sets S(a),

(ST) ReK(z,w) d/-l(z) d/-l(w) ~ C(/-I)/-I(S(a» , S(a) S(a)

(with C(/-I) independent of a in Bn+l)'

Here C(p.) denotes positive constants, possibly with different value at each occurrence.

The conditions (SC) is obtained by testing the boundedness of J, the inclusion ofDA into L2 (d/-l), on a localized bump. The condition (ST) is obtained by testing the boundedness of the adjoint, J*, on a localized bump. Hence the third statement of the theorem is very similar to the hypotheses in some versions of the T(l) theorem. 'Ibis viewpoint is developed in [17].

In light of (2) we had used ReK(z,w) in analyzing Carleson measures. When estimating the size of Re K(z, w) in the tree model it was useful to split the tree into equivalence classes and use the geometry of the quotient structure. That is the source of the name "split-tree condition" for (ST). Versions of such a quotient structure will be considered in the later part of this paper.

Problem 1. Theorem 1 gives a geometric characterization of the multiplier norm for fixed n, but we do not know how the relationship between the different constants C p.), and between them the multiplier norm of g, depend on the dimension. Good control of the dependence of the constants on the dimension would open the possibility of passing to the limit as n -+ 00 and providing a characterization of the multiplier norlll for the infinite-dimensional DA space.

An alternative approach to the characterization of the Carleson measures is in {17], where Tchoundja exploits the observation made in [3] that, by general Hilbert space theory, the inequality in (2) is equivalent (with a different C(/-I» to

2

(1.3) ReK(z,w)cp(z) d/-l(z) d/-l(w) ~ C(/-I) BO>+I B .. +1

We mention here that (1.3) is never really used in [3], while it is central in [17]. Tchollndja's viewpoint is that (1.3) is the L2 inequality for the "singular" integral having kernel Re K(z, w), with respect to the non-doubling measure /-I. He uses the fact ReK(z,w) ~ 0 to insure that a generalized "Menger curvature" is positive. With this in hand he adapts some of the methods employed in the solution of the Painleve problem to obtain his proof. His theorem reads as follows.

Theorem 2. A meosure /-I on Bn+l is Carles on for DA if and only if any of the following (hence, all) holds.


(1) For some 1 < p < 00,

In+l (In+l ReK(Z,W)'P(Z)dJL(Z)Y dJL(w) ~ C(JL} 1,,+1 r.p1'p.

(2) The inequality in (1) holds for all 1 < p < 00.

(3) The measure JL satisfies the simple condition (SC) and also for some 1 < p < 00 the inequality

(1.4) r (r Re K(z, w) dJL(W))P dJL(z} ~ C(J.&}J.&(S a ). JS(a) iSla)

(4) Condition (3) holds for aliI < p < 00.

(Actually [17] focuses on the p ~ 2 but self adjoint ness and duality then gh1! the expanded range.) Observe that, as a consequence of Theorems 1 and 2 the condrti n (1.4) equivalently holds for some 1 < p < 00 then it holds all 1 < p < 00. On the other hand, it is immediate from Jensen's inequality that if the inequality h Ids for some p then it holds for any smaller pj hence the condition in Theorem 1, ST), IS

a priori the weakest such condition.

Problem 2. "Which geometric-measure theoretic properties foll w fr m the fact that the Carleson measures for the DA space satisfy such "reverse HIder inequalities" ?

Indeed, the same question might be asked for the Carleson measures for a variety of weighted Dirichlet spaces, to which our and Tchoundja's methods apply It is interesting to observe that, while our approach is different in the DA case and in other weighted Dirichlet spaces (see [31 and the references quoted there ; Tchoundja's method works the same way in both cases. On the other hand, hIS proof does not encompass (ST) in Theorem 1.

We conclude this introduction with an overview of the article. Changing coordinates by stereographic projection, we see in Section 2.1 that

on the Siegel domain (generalized upper half-plane)

Un+l = {z = (z',Zn+1) E en x e: Im(Zn+1) > z' 2}

K is conjugate to a natural kernel H 1

H(z, w) = -. -_-------= l(Wn +1 - Zn+1) 2 - z'· w'

This is best seen changing to Heisenberg coordina.tes:

[(, tj r1 = [z', Re(zn+l); Im(Zn+1) - \z'\21. The Heisenberg group IHIn has elements [(, t1 E en x lR and group law [(, t1'

[~, s1 = [( +~, t + s + 2 Im«· e)1. The kernel can now be written as a convolution kernel: writing

we have H([(,tjr], [~,sjr]) = 2'Pr+p([~,s1-1. [(,s])

Because of the connection with the characteriza.tion of the multipliers for DA our main interest is in Re (H (z, w) ). The numera.tor a.nd the denominator of Re( ';1')


each have an interpretation on terms of the sub-Riemannian geometry of lHln. The denominator is the Koranyi distance to the origin, at scale ..;r, while the nwnerator is the Koranyi distance from the center of the group lHln to its coset passing through [(, t], again at the scale ..;r. We see, then, that the kernel /pr reBects the two-step stratification of the Lie algebra of lHln.

The Heisenberg group, which has a dilation as well as a translation structure, can be easily discretized, uniformly at each scalej and this is equivalent to 8 discretization of Whitney type for the Siegel domain Un+!' The dyadic boxes are fractals, but in Section 2.2 we see that they behave sufficiently nicely for us to use them the same way one uses dyadic boxes in real upper-half spaces. The same way the discretization of the upper half space can be thought of in terms of a tree, the discretization of the Siegel domain can be thought of in terms of a quotient structure of trees, which is a discretized version of the two-step structure of the Heisenberg Lie algebra.

In Section 3, we see how the DA kernel (rather, its real part) has a natural discrete analog living on the quotient structure. We show that, although the new kernel is not a complete NevsnJinna- Pick, it is nonetheless a positive definite kernel. In [3], the analysis of a variant of that discrete kernel led to the characterization of the multipliers for DA. We do not know if an analogous fact is true here, if the discrete kernel we introduce contajns all the important information about the kernel H.

We conclude by observing, in Section 4, that, as a consequence of its "conformal invariance," a well-known kernel on the tree, which can be seen as the discretization of the kernel for a weighted Dirichlet space in the unit disc, has the complete Nevanlinna- Pick property.

Notation. Given two positive quantities A and B, depending on parameters ct, {J, .•• , we write A :::::: B if there are positive c, C > 0, independent of 0, {J, . .. , such that cA ::; B ::; C A.

2. A flat version of DAd

2.1. From the ball to Siegel's domain. In this section, we apply stereographic projection to the DAd kernel and we see that it is conjugate to a natural kernel on the Siegel domain. In this "Bat" environment it is easier to see how the D~ kernel is related to Bergman, and hence also to sub-Riemannian geometry. A discretized version of the kernel, analogous to the dyadic versions of the Hardy space kernel in one complex variable immediately comes to mind.

We follow here the exposition in [15]. As we mentioned, Siegel's domain Un+1

is defined as

Un+l = {z = (Zl, ... , Zn+1) = (z', zn+1) E en+! : 1m Zn+1 > Iz'12}.

Forz,w inUn+l, define • 1

r(z,w) = "2(wn+l - Zn+d - z' . w'.

Consider the kel'nel H: Un+! X Un+l -+ e,

(2.1) H(z,w):::; 1 r(z,w)


Proposition 1. The kernel H is conjugate to the Drury Aroe8on kernel K. Hence, it is a definite positive, (universal) N evanlinna - Pick kernel.

In fact, there is a map 4»: lBn+1 ~ Un +1 such that:

(2.2)

PROOF. Let lBn+1 be the unit ball of en+1 and let Un +1 be Siegel's d mam.. There is a biholomorphic map z = 4»«) from lBn+1 onto Un+1:

{

zn+1 = iG ~ ~:::) (k

Zk = , if 1 ~ k ~ n, 1 +(n+1

having inverse

{

<n+1 = ~ ~ :::: /" 2izk ."k = . ,if 1 ~ k ~ n.

1+ Zn+1

Equation (2.2) follows by straightforward calculation. 0

Remark 1. The map f f--t j, j(z) = 2/(i + Zn+1 f 4»-1 z ,is an isometry from the Hilbert space with reproducing kernel K to the HUbert space v.;th reproducing kernel H. We call the latter DAu.

Problem 3. Find an interpretation of the DAu norm in terms of WeIghted Dirichlet spaces on Un +1.

Recall (see [3]) that a positive measure I-' on lBn+1 is a Carleson measure for DA if the inequality

(2.3)

holds independently of f. The least constant \ I-' eM DA = G 1-') for which 2.3 holds is the Carleson measure norm of 1-'.

The following proposition is in (31.

Proposition 2. The Carleson norm of a measure I-' on Bn+l IS comparable with the least constant G1 (I-') for which the inequality below hold for all measurable 9 ~ 0 on lBn+b

r r Re(K(z,w))g(z) dl-'(z)g(w) dl-'(w) ~ G1 (1-') r g2dl-" 18,,+1 1B"+l 18,,+1

As a corollary, we obtain the following.

Theorem 3. Let I-' ~ 0 be a measure on Bn+l and define its normalized pull-back on Un+ 11

djL(z):= \i+Zn+d2dl-'(4)>-1(z)) Then, I-' E GM(DA) if and only if jL satisfies

r 1 Re(H(zlw))g(z)djL(z)g(w)djL(w) ~ G2(jL) 1 g2djL. 1U"+1 U"+l U,,+l

Moreover, G(I-') :::: G2(jL)·

TWO VARIATIONS ON THE DRURY ARVESON SPACE 41

Problem 4. Find a natural, operator-theoretic interpretation for H; in analogy with the interpretation of K in [11].

The kernel H is best understood after changing to Heisenberg coordinates which help reveal its algebraic and geometric structure. For z in Un+l, set

z = (z',Zn+l) == [(,tiT]:= [z',Rezn+l;Imzn+l-\z'\2].

The map z ~ [(, tj r] identifies Un+l with ]R2n+2, and its boundary 8Un+l with ]R2n+l. In the new coordinates it is easier to write down the equations of some special families of biholomorphisms of Un +1:

(i) rotations: RA: [(,tjr]1-t [A(,t;r], where A E SU(n); (ii) dilations: Dp: [(, t; rjl-t [p(, P2ti p2r); and (iii) translations: Tt{,t): [{,Sjpjl-t [(+e,t+s+2Im«(·e);pj·

This Lie group of the translation is the Heisenberg group JH[n == ]R2n+l which can be identified with 8Un+l- The group operation is

[(, t]· [{, sj = [( + e, t + 8 + 2 Im«(· e)j

and thus Tt{,t): [{,SiP] ~ [[(,t]. [e,8jipj. We can foliate Un+l = Up>o JH[n(p), where JH[n(p) = {[(, tiPI : [(, tj E JH[n} is the

orb1t of [0,0; p] under the action of JH[n. The dilations D p on Un+l induce dilations on the Heisenberg group:

8p [(, tj := [p(, p2tj. The relationship between dilations on JH[n and on Un+l can be seen as action

on the leaves:

Dp: IHI"(r) -+ IHI"(p2r), Dp[(, tj rl = [8p[(, tlj p2rl.

The zero of the group is ° = [0,0] and the inverse element of [(, tj is [-(, -t]. The Haar measure on lHIn is d( dt. We let d(3 to be the measure induced by the

Haar measure on 8Un+l: d(3(z) = d( dt.

We also have that dz = d( dt dr is the Lebesgue measure in Un+l'

We now change H to Heisenberg coordinates.

Proposition 3. If z = [(, tj r] and w = fe, 8j p], then

24

where

) r+p+le-(12-i(t-s-2Im(e·())

H(z,w = 2· _ 2 (r+p+\e-(\2)2+ (t-s-2Im(e·())

= 2r,or+v([e, 8]-1. [(, t]),

.,.. + \(12 - it

r,or([(, t]) = (r + \(12)2 + t2

The expression in Proposition 3 is interesting for both algebraic and geometric rE'.aJ30ns. Algebraically we see that H can be viewed as a convolution operator. From a geometric viewpoint we note that the quantity 1I[(,tlll := (t2 + 1(14)1/4 ie the Koranyi no1111 ofthe point [(, t) in IHln. The distance associated with the norm ie

dy,,([(, t], [{, s]) := \lle, st1[(, tll\·

Hence, the denominator of r,or might be viewed as the 4th power of the Korany: norm of [(. t) "at the scale" '1'1/2.

48 N. ARCOZZl ET AL.

In order to give an intrinsic interpretation of the numerator, consider the center T = {[O, t} : t E JR} oflllIn, and the projection II: lHln -+ en == lHln /T: II(l(, t} = <. Then, independently of t E JR,

\(\ = dHn (T, l(, tl· T)

is the Koranyi distance between the center and its left (hence, right) translate by {(, t}. The real part of the DA kernel has a twofold geometric nature: the denominator is purely metric, while the numerator depends on the "quotient structure" induced by the stratification of the Lie algebra of lHln. This duality is ultimately responsible for the difficulty of characterizing the Carleson measures for DA.

The boundary values of Re('Pr),

\Z\2 (2.5) 'Po(l(, tD := \z\4 + t2'

were considered in {121 (see condition (1.17) on the potential) in connection with the Schrodinger equation and the uncertainty principle in the Heisenberg group.

Problem 5. Explore the connections, if there are any, between the DA space, the uncertainty principle on lHln and the sub-Riemannian geometry of H".

We mention here that, at least when n = 1, the kernel <Po in 2.5 satisfies the following, geometric looking differential equation:

1 0 Do \HI 'Po ([', tD = "2 0\(\2 <Po(l', t1),

where DoH = XX + YY is the sub-Riemannian Laplacian on l8L Here, with' = x + iy E C, X and Y are the left invariant fields X = oz + 2y8t and Y = 8 - 2xBt. See [81 for a comprehensive introduction to analysis and PDEs in Lie groups with a sub-Riemannian structure.

2.2. Discretizing Siegel. The space Un +l admits a dyadic decomposition, which we get from a well-known [161 dyadic multidecomposition of the Heisenberg group, which is well explained in [181. We might get a similar, less explicit decomposition by means of the general construction in [10].

Theorem 4. Let b ;::: 2n+ 1 be a fixed odd integer. Then, there exists a compact subset To in JH[n such that: To is the closure of its interior and II(To) = [-!, ~l2ni

(1) m(8To) = 0, the boundary has null measure in IHr'; (2) there are b2n+2 affine maps (compositions of dilations and translations

.Ak o/lHIn such that: To = Uk .Ak(To) and the interiors of the sets .Ar.(To are disjoint;

(3) the sets n (.Ak (To») divide [-i, i12n into b2n cubes with disjoint interiors.

each such cube being the projection of b2 sets Ak (To).

Consider now Un+l, let b be fixed and let mE Z. For each k = (k',k2n+l) E z2n X Z, consider the cubes

Qr = c5b- m 7k(To) X [b-2m-2, b-2ml Qm lb-2m-2 b-2ml Qm U = Ii: x , = k C n+l,

with Qr:: c lHln. Let T(m) be the sets of such cubes, u(m) the set of their projections, and T = UmEZ T(m) J U = UmEZ u(m). We say that eo cube Q' in T(m+l)

(respectively u(m+l») is the child of a cube Q in T(m) (respectively u(m)}, if Q' c: Q.


In order to simplify notation, if Q is a cube in T, we write [Q] = Il(Q). We state some useful consequences Theorem 4.

49

Proposition 4. (i) Each cube in T(rn) has b2n+2 children in T(rn+1) and one parent in T(rn-l); hence, T is a (connected) homogeneous tree of degree b2n+2 •

(ii) Each cube in U<m) has b2n children in u(m+1) and one parent in U(m-l); hence, U is a (connected) homogeneotLS tree of degree b2n •

(iii) For each cube Q in T(m), there are Koranyi balls B(zQ, eob-m) and B(wQ,clb-m) in lHI'" such that

B{wQ,c1b-m) X [b 2m 2,b-2m] C Q C B(WQ,C2b- m) X [b-2rn-2,b-2m].

We say two cubes Ql, Q2 in T are graph related if they are joined by an edge of the T, or if they belong to the same T( rn) and there are points Zl E Q 1, Z2 E Q2 such that dH"(ZllZ2)::; b-m • An analogous definition is given for the points of U. We consider on T the edge-counting distance: d(Ql, Q2) is the minimum number of edges in a path going from Ql to Q2 following the edges of T: the distance is obviously realized by 8. unique geodesic. We also consider a graph distance, do Qll Q2) ::; d(Qll Q2), in which the paths have to follow edges of the graph G just defined. The edge counting distance on the graph is realized by geodesics, but they are not necessarily unique anymore. Similarly, we define counting distances for the tree and graph structures on U.

Given 8. cube Q in T, define its predecessor set in T, P( Q) = {Q' E T : Q ~ Q'}, and its graph-predecessor set PG(Q) ={ Q' : dG(Q', P(Q)) ::; 1}. We define the level of the confluent of Ql and Q2 in G as

2.6 d(Ql A Q2) := max{d(Q) : Q E PG(Ql) n PG(Q2)}.

We don't need, and hence don't define, the confluent Ql A Q2 itself.) Similarly, we define predecessor sets in T and G for the elements of U, and the

level of the confluent in the graph structure, using the same notation. Observe that P [Q1 = [P(Q)] := ([Q'] : Q' E P(Q)}, P([Ql) := [P(Q)], but that the inequality

d(Ql A Q2) ::; d([Ql] A [Q2])

cannot in general be reversed.

Theorem 5. Let z := [(, tir1, w = Ie, SiP] be points in the Siegel domain Un+!. and let Q(z), Q(w) be the cubes in T which contain z, w, respectively (if Z is contained m more than one cube, we choose one of them). Then,

(2.1) bd(Q(z)i\Q(w» ~ (r + P + I( - e12)2 + (t - s - 2Im«(. {))2)1/4

18 appronmately the ! -power of the denominator of H (z, w). On the other hand,

(2.8) bd([Q(z)Ii\[Q(w)]) ~ (r + P + I( _ eI2 )1/2

18 approximately the ~-power of the numerator of Re H(z, w). We have then the equivalence of kernels:

(2.9) Re H(z, w) ~ b2d(Q(.a:)i\Q(w))-d([Q(.a:)]i\[Q(w)])

Thus we have modeled the continuous kernel by 8. discrete kernel. This kernel, however, lives on the graph G, rather than on the tree T.

Theorem 5 allOW8 a discretization of the Carleson measures problem for the DA space on Un+1- Given a measure /i. on Un +1. define a measure IJ~ on the graph G:


/Ld(Q) := JQdji. Then, ji satisfies the inequality in Theorem 3 if and only if ",I is such that the inequality

(2.10) L: L: I?d(qi\q')-d([q]i\[q']\o(q)",d(q)cp(q')",~(q') ::; C ",D) L:c1",' qEGq'EG G

holds whenever cp ~ 0 is a positive function on the graph G. In the Dirichlet case, inequality (2.10) is equivalent to its analog on the tree.

Given q,q' in T, let qAq' be the element p contained in [o,q]n[o,q'} for whichd(p IS

maximal. An analogous definition can be given for elements in U. The tree-analog of (2.10) is:

(2.11) L: L: b2d(qAq')-d([q]A[q'Dcp(q)",d(q)cp(q')",D(q')::; C ",' L~I£'· qETq'ET T

Problem 6. Is it true that the measures ",D such that 2.10 h Ids for all cp: T -+ [0,+00), are the same such that (2.11) holds for all cp: T -+ [0,+00 .

There is a rich literature on the interplay of weighted inequalities, Carleson measures, potential theory and boundary values of holomorphic functi ns.

Problem 7. Is there a ''potential theory" associated with the kernel R.e H giving, e.g., sharp information about the boundary behavior of functions in D'\?

Before we proceed, we summarize the zoo of distances usually employed in the study of Un+1 and of lHln as a guide to defining useful distances on T and U. We have already met the Koranyi distance lI[e,s]-l. [(,tl between the points [(,t and [e, s] in lHln. The Koranyi distance is bi-Lipschitz equivalent to the CarnotCaratheodory distance on lHln. We refer the reader to 8] f r a thorough treatment of sub-Riemannian distances in Lie groups and their use in analysis. The point we want to stress here is that the Carnot - Caratheodory distance is a length-distance, hence we can talk about approximate geodesics for the Koranyi distance itself.

Although it is not central to our story, for comparison we recall the Bergman metric (3 on Un+1 • It is a Riemannian metric which is invariant under Heisenberg translations, dilations and rotations. Define the I-form w([(, t]) = dt - 21m (.d( . Then,

(2.12) d{3([(, t; r])2 = \d(\2 + w([(, t]): + dr2

•

r r

This can be compared with the familiar Bergman hyperbolic metric in lBn+1•

2 \dz\2 \z . dz 2 df3Bn +1 (z) = 1 _\Z\2 + (I_\z\2)2'

Lemma 1. (i) For each r > 0, consider on JIr(r) the Riemannian d\S-tance d{3~ obtained by restricting the two-form d{32 to lHln(r). Then, the following quantities are equivalent for \I [(, tll\ ? ..;r:

((\(\2 + r)2 + t2) 1/4 ~ \I [(, t] \I ~ vr{3r([(, t; r], [0, 0; rD.

(ii) A similar relation holds for cosets of the center. Let [T; r] = T· [0,0; rl be the orbit of [0,0; rJ under the action of the center T. Then,

(\(12 + r )1/2 ~ den ([(, t] . T, T) ~ vr{3r ([[(, tl· T; r 1, [T; rn


PROOF OF LEMMA 1. The first approxima.te equality in (i) is obvious. For the second one, using dilation invariance

';;:13 ({(, tj r),{O, OJ r]) = ';;:13

Dyr({O,O; 1]) =';;:13

( t Dyr r.;' -j 1 ,

vr r

( t ..ft,-;I,[O,O;I)

r r

Since the metrics 13 and dH" define the same topology on IHln(r), the last quantity is comparable to y'rdHn ([(/..ft, tlr; 1], [0, OJ 1]) when 1 S II (/..ft, tlrjll $ 2, by compactness of the unit ball with respect to the metric and Weierstrass' theorem. Since the metric I3r is a length metric and dllln is bi-Lipschitz equivalent to a. length metric (the Ca.rnot CaratModory distance), then, when 1 $ 1I[(/..ft,t/rlll, we have

( t ..ft' - , [0, OJ r r

= dlHln ([(, t), [0,0]) . r r

The proof of (ii) is analogous. o PROOF OF THEOREM 5. We prove (2.7), the other case being similar (easier,

in fact. Suppose that d(Q(z),Q(w)) = m. Then, d(Q(z)),d(Q(w)) $ m, hence,

b-m :s .;r, v'P and there are Q1, Q, Q2 in T(m) such that Q(z) ~ Q1 G Q G Q2 S Q W • We have then that

b-m ~ ma.x{ vir. yp, cdRn ([(, t), [~, sj)} ~ ((r+p+l( _~12)2+ (t-s-2 Im((.~))2) 1/4 ;

the left-band side of (2.7) dominates the right-hand side. To show the opposite inequality, consider two cases. Suppose first that ..ft ~

.jP,du {(.t],[~.s]) and that b-m ~ .;r ~ b-m- 1. Then, Q(z) G Q(w). Hence,

m$d(Q z)ii.Q(w)) $m+l and

b-d(Q .z: AQ(w» ~ b-m ;:: ((r + p + I( _ ~12)2 + (t _ 8 _ 2Im(( . ~))2) 1/4.

Suppose now that dHn({(,t],{~,s]) ~ .;r,.jP and choose m with m $ d(Q(z) A Q w) S m+ 1. Let Qm(z) and Qm(w) be the predecessors of Q(z) and Q(w) in Tm we lise here that d(Q(z)),d(Q(w)) ~ m). Then, Qm(z) G Qm(w), hence

dB 1(. t]. {~, sD :s b-m:

b-d(Q(.z:)AQ(w» ~ b-m ;:: dllln ({(, tj, {~, s])

~ ((r + p + I( - e12)2 + (t _ 8 - 2 Im(( . ~))2)1/4.

The theorem is proved. o It can be proved that

1 + do(Q(z),Q(w)) ~ 1 + l3(z,w),

where fJ is the Bergman metric and do is the edge-counting metric in G. The for the kernel R.e H in Theorem 5 reflects the graph structure

of the set of dyadic boxes. We might define a new kernel using the tree structure only 88 follows. Given cubes Q11 Q2 in T, let

Q1 A Q2 := ma.x{Q E T: Q $ Q1 and Q2 $ Q2}


be the element in T such that [0, Qd n [0, Q2] = [0, QI "Q2]' Define similarly [QI] " [Q2] in the quotient tree U. Define the kernel:

HT(Z, w) := b2d( Q(%)AQ(UI»)-d([Q1IA[Q2)), Z, wE lin+l

As in Theorem 5, there is a slight ambiguity due to the fact that there are several Q's in T such that Z E Q. This ambiguity might be removed altogether by distributing the boundary of the dyadic boxes among the sets sharing it.

Because nearby boxes in a box can be far away in the tree, it is not hard to see that HT is not pointwise equivalent to R.eH. However, when discretizing the reproducing kernel of Dirichlet and related spaces the Carleson measure inequalities are the same for the tree and for the graph structure. We don't know if that holds here. See [6] for a general discussion of this matter.

In the next section, we discuss in greater depth the kernel HT.

3. The discrete DA kernel

Here, for simplicity, we consider a rooted tree which we informally view as discrete models for the unit ball. The analogous model for the upper half space would have the root "at infinity."

Let T = (V (T), E(T)) be a tree: V (T) = T is the set of vertices and E T) is set of edges. We denote by d the natural edge-counting distance on T and,for %, YET, we write [x, y] for the geodesic joining x and y. Let 0 E T be a distinguished element on it, the root. The choice of 0 induces on T a level structure: d = do: T ~ N, x 1-4 d(x, 0). Let (T,o) and (U,p) be rooted trees. We will use the standard notation for trees, x 1\ y, x ~ y, x-1 , C(x) for the parent and children of x, P x and S x for the predecessor and successor regions. Also recall that for f a function on the tree the operators 1 and I* produce the new functions

1f(x) = ~ fey); r I(x) = L fey)· IIEP(x) yES(",)

A morphism of trees q,: T --+ U is a couple of maps q,v: T --+ U, q, E: E T) -4

E(U), which preserve the tree structure: if (x, y) is an edge of T, then q,E(X,y) = (q,v(x),q,v(y)) is an edge of U. A morphism of rooted trees q,: (T,o) --+ (U,p is a morphism of trees which preserves the level structure:

dp(q,(x)) = do(x).

The morphism q, is an epimorphism if ~v is surjective: any edge in U is the image of an edge in T.

We adopt the following notation. If x E T, we denote [xl = q,v(x). We use the same symbol 1\ for the confluent in T (with respect to the root 0) and in U (with respect to the root p = [0]).

A quotient structure on (T,o) is an epimorphism ~: (T,o) --+ (U,p). The rooted tree (U,p) was called the tree of rings in [3].

Recall that b ~ 2n + I is a fixed odd integer. Fix a positive integer N and let T be a tree with root 0, whose elements at level m ~ I are ordered m-tuples a = (ala2 ... am), with aj E Z"N+l, the cyclic group of order bN +1. The children of a are the (m + I)-tuples (ala2 ... amo), a E Z"N+l, and the root is identified with a O-tuple, so that each element in T has bN +1 children. The tree U is defined similarly, with bN instead of bN +1.


Consider now the group homomorphism i from Zb to ZbN+2 given by i([k]mod b)

= [hN klmod bN+1 and the induced short exact sequence

The projection IT induces a map II>v: T -+ U on the set of vertices,

II>v(ala2 .. . am) := (II>v(al)lI>v(a2)" . II> v (am)) ,

which clearly induces a tree epimorphism 11>: T -+ U. Here a way to picture the map 11>. We think of the elements C of U as "boxes" containing those elements x in T such that [xl := lI>(x) = C. Each box C has b N children at the next level, CI, ... , CbN. Now, each x has b N +1 children at the same level, b of them falling in each of the boxes CJ'

We t.hink of the quotient structure (T, U) as a discretization of the Siegel domain Un+l' with b = b2 and N = n.

The discrete DAN +l kernel K: TxT -+ [0, 00) is defined by

K(x, y) = b2d(:z:"y)-d([:z:],,[y])

Note that it is modeled on the approximate expression in (2.9).

Theorem 6. The kernel K is positive definite. In fact,

L b2d Zl\y)-d :z:)"[v]) p,(x)p,(y)

Z vET

= I*p,o 2+b-1"'\1*p,(z)\2+~ '" b 2d(z"w)-d([%),,[w])\1*p,(z)_1*p,(w)\2 b ~ 2 ~ . z-j.o z-j.wET

[z]=[w]

The theorem will follow from the following lemma and easy counting.

Lemma 2 (SuJllmation by parts). Let K: TxT -+ C be a kernel on T, having the f07m K(x,y) = H(x A y, [x] A [y]) for some function H: T x U -+ C. Then, 'f p,: T -+ C is a function having finite support,

3.1 L K(x, y)p,(x)p,(y) = H(o, [0])\1* p,(0)\2 %v

+ L IH(z A w, [z] A Iw]) - H(Z-l Aw-l, Iz-1] A Iw-1])]I" p,(z)I°p,(w) % wET {a}

[z]=[w)

PROOF. Let Q be the left-hand side of (3.1). Then,

Q = L H(x A y, Ix] A Iy])p,(x)p,(y)

= L L H(z A w, C) L p, (x) p,(y) GEU z,wET :z:~%,v~1U

Izl"lvl=G

= L L H(ZAw,C)A(z,w), GEU z,wET


If z -:j:. w,

A(z, w) = 2: p,(x)p,(y) :r:~Z,II2:w [:r:]1\[1I]=0 2: 2: /*p,(s)/*p,(t) + p,(z)(/*p(w) - p,(w)}

DtfFEU sEC(%),[s]=D D,FEC(O) tEC(W),[t]=F

+ (/*p,(z) - p(z»p(w) + '" z)", w .

On the other hand,

r p,(z)[* p,(w) = p,(z) (I* p,( w) - .u( w» + (1* p,(z) - .u(z »p,(w) + p(z)p w)

+ L L r p, s I·", W •

D,FEC(O) [s]=D,sEC % [t]=F,tEC w

Hence, if z =F w,

A(z,w) = 1*1J-(z)I*",(w) - L L: 1*p s]*p t FEC(C) [s]=D,sEC %)

[t]=F,tEC(w

= /*p,(z)/*IJ-(w) - L L: J* I-'(s) L: I*p t . FEC(C) [s]=D, [t =F,

sEC(.z:) tEC w

In the case of equality,

A(z, z) = L IJ-(X)p,(y) :r:,y~%

[:r:]I\[Y]=O -,-----,--~ = IJ-(Z) (l*IJ-(Z) -IJ-(Z» + IJ-(Z) (1* p(z) - p(z» + p z) 2

+ L: L: J*p(s)F;t. D"#F [s)=D,sEC(z)

D,FEC(C) [t =F,tEC(w

On the other hand,

JI*p.(zW = p.(z)""'(I-*IJ--(z-)---p.-(z-») + p.(z)(/*",(z) - p(z» + \",(z)\2

+ \ L L I*IJ-(S)\2 DEC(O) [s)=D,

sEC(z)

= ",(z) (I*",(z) - p.(z») + p.(z)(I*",(z) - I-'(z» + 114(z)12

Comparing:

(3.2)

+ L \ L 1*14(8)\2 + L L /* IJ-(S)y.;(ij .. DEe(D) [s]=D, D"#F [s]=D, sEC(z)

BEC(z) D,EEC(C) [t)=F, tEC(w)

A(z,z) = \J*1J-(z)l2 - L \ L I*IJ-(s)\2. DEC(D) [s) D

"EC(")


Then,

Q= 2.: LH(z/\w,C) I*JJ(z)I*~(w)- 2.: LI*JJ(S) 2.: 1*JJ(t) CEU(z)=~EC DEC (C) (a]-D (t]=F

aEC(z) tEC(w)

= L [H(z/\w,C)-H(z-l/\W \C 1)]I*JJ(z)I*JJ(w) + H(o, [0]) iI*JJ(O) 12 ,

1.1]-[,"]=0 d(z)-d(tu)~1

which is the desired expression. In the last member of the chain of equalities,we have taken into account that

term rJJ(z)I*JJ(w) appears twice in the preceding member (except for the root term). 0

PROOF OF THEOREM 6. Let Q be the left-hand side of (3.1). By Lemma 2,

Q = 1*JJ(O) 2

+ L: [b2d(z"tu)-d([z]l\[tu]) - b2d( .. -1I\tu- 1)_d([ .. -1)I\[w- 1])] r JJ(z)l",.,.( w)

.. ,WET {o} [z = wJ

\,re have two consider two cases. If z =F w, then z /\ w = z-1 /\ w-1, [Z-1] /\ [w-1] = [z] /\ w1)-I, so that the corresponding part of the sum is

3.3 Q1 = -(b - 1) 2.: b2d(zl\w)-d([zJI\[w]) r ,.,.(z )1*,.,.( w) zi'wET\{o}

( .. ]=[w]

IT z = w, then z-1 /\ z-1 = z-1, hence the remaining summands add up to

The tell!! Ql in (3.3) contains the mixed products of

zi'wET\{o} (z]=[.o]

= Ql + (b -1) LiI*,.,.(zW 2.: b 2d( .. l\w)-d([z]l\[w]) ,

zi'o w: [w]=[z]

The last 811m can be computed, taking into account that, for 1 ::; k ::; d(z), there are (b - 1)bk - 1 W'B for which [w] = [z] and

d(z) = d([z] /\ [wD = d(z /\ w) + k,

by the special nature of ~: T -+ U:

d( .. )

L: b 2d( .. I\.o)-d«(z)I\(.o)) = 2.:(b - 1)bk - 1b2(d(z)-k) d(z)

w;(wl=(z] k=l d(z)

= (b - l)bd(")2.:2- k - 1 = .!..(bd(z) - 1) k=1 b


Hence,

R = Ql + Q2 - b ~ 1 ~)l*IL(ZW = Q -\1"IL(O) 2 _ b ~ 1 L l*IL(Z) 2,

z~o z~o

as wished. o

Problem 8. The discrete DA kernel in Theorem 6 does not have the complete Nevanlinna-Pick property. This is probably due to the fact that the kernel is a discretization of the real part of the DA kernel on the unit ball, not of the whole kernel. Is there a natural kernel on the quotient structure i{I: T ~ U which is complete Nevanlinna- Pick?

In the next section, we exhibit a real valued, complete Nevanlinna-Pick kernel on trees.

4. Complete Nevanlinna-Pick kernels on trees

Let T be a tree: a loopless, connected graph, which we identify with the set of its vertices. Consider a root 0 in T and define a partial order having 0 as minimal element: x ::; Y if x E [0, yj belongs to the unique nonintersecting path joinin., and Y following the edges of T. Given x in T, let d(x) := tt[o,x] -1 be the number of edges one needs to cross to go from 0 to x. Define x A y =: max[o,x] n [o,y to be the confluent of x and y in T, with respect to o. Given a summable function IL: T -t IC, let 1* IL(X) = Ey~x IL(Y)·

Theorem 7. Let A > 1. The kernel

K(x,y) = Ad(ZAy)

is a complete Nevanlinna - Pick kernel.

Our primary experience with these kernels is for 1 < A < 2. At the level of the metaphors we have been using, 2d(ZAy) models IK(x, y) for the kernel K of (1.1). We noted earlier that the real part of that kernel plays an important role in studying Carleson measures. For that particular kernel passage from ReK to K loses a great deal of information. However in the range 1 < A < 2 the situation is different. In that range Ad(xl\y) models IKQI, 0 < Q < 1 and the KQ are the kernels for Besov spaces between the DA space and Dirichlet spaces. For those kernels we have IKQI ~ ReKQ making the model kernels quite useful, for instance in [5].

These kernels also arise in other contexts and the fact that they are positive definite has been noted earlier, [13, Lemma. 1.2; 14, (1.4)).

We need two simple lemmas.

Lemma 8 (Summation by parts). Let h,IL: T ~ C be functions and let M = 1* IL. Then,

Eh(x I\Y)IL(X)IL(Y) = h(o)\M(0)12 + L [h(t) - h(t-1)lIM(t)\2. :r.1I tET\{o}


PROOF.

= L h(t) L p(X)~(Y) , ZI\!I=t

= L h(t) Ip(t)12 + p(t)(M(t) -It(t)) + It(t)(M(t) -It(t)) + L M(z)M(w) t z#w;z,w>t;

= Lh(t) IM(tW-, L IM(z)12 , a>t

d(z,t)=l

d(w,t)=d(z,t)=l

which is the quantity on the right-hand side of the statement. o 4. Fix a new root a in T and let da and I\a be the objects related to

tins new root. Then,

da(x 1\.0 y) = d(x I\. y) + d(a) - d(x 1\ a) - d(a 1\ y).

PROOF. The proof is clear after making sketches for the various cases. 0

PROOF OF THE THEOREM 7. The kernel K is complete NevaJJli.nna-Pick if end only if each matrix

A = 1- K(Xi,XN)K(XN,Xj) 4.1 K(XN' xN)K(Xi' Xj) i,j=l...N-l

is positive definite for each choice of Xl,"" XN in Tj see [2]. Let a = XN. The (i,j)th entry of A is, by the second lemma,

A -do % 1\% • By the first lemma, A is positive definite.

References

1-o

1. J Agler and J. E. McCarthy, Oomplete Ne'llanlinno - Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111 124.

2 , PICk \nterpolatl.on and Hilbert function spaces, Grad. Stud. Math., vol. 44, Amer. Math. Soc., Providence, RI, 2002.

3. N. Aroozzi, R. Rochberg, and E. Sawyer, Oarleson measures/or the Drury-Arveson Hardy 6pace and other Bes01J-Sobole'll spaces on complex balls, Adv. Math. 218 (2008), no. 4, 1107-1180.

4. , The dwmeter space, II restriction of the Drury-Arveson-Hardy space, Function Spaces, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 21-42.

5. • Carleson measures and interpolating sequences for Beso'll spaces on compie:D balls, Melli. Amer. Math. Soc. 182 (2006), no. 859.

6 N. ArcozZl, R. Rochberg, E. Sawyer, and B. Wick, Potential theory and Oarleson measures on "'''cell and grophs, in preparation.

7 W. An'eson, Subalgebt'Q./I 0/ C· -algebt'Q./I. III: Multl.'IIariable operotor theory, Acta Math. 181 1998). no. 2, 159 - 228.

8. A. BonfigJioJi, E. Lanconelli, and F. Uguzzoni, Strotljied Lle groups and potential theory for the\r sub-Laplacw.ns, Springer Monogr. Math., Springer, Berlin, 2007.

9. Z. Chen, Charoctel\zatwns 0/ Arveson', Hardy 'pace, Complex Var. Theory Appl. 48 (2003), no. 5,453-465.

10. M. Christ, A T( b) ""th ."marks on analyt,c cq,pacity and the Oauchy integral, Colloq. Math, 60/61 (1990), no. 2, 601 628.

11 S. W Drury, A general,zatwn of von Neumann', inequalitll to the complex bali, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300 304.


12. N. Garofalo and E. Lanconelli, Frequency functions on the Het8enberg group, the uncertamtll principle end umque contmuatton, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313-356

13. U. Haagerup, An e:wmple of e nonnucleer C·-algebra, whu:h has the metru: eppnna~ property, Invent. Math. 50 (1978/79), no. 3, 279-293.

14. Yu. A. Neretin, Groups of h~erarch.omorph'l8mB of trees and related Hdbert spaces, J Funct.. Anal. 200 (2003), no. 2, 505-535.

15. E. M. Stein, Hermonic enclysia: reel-venable methods, orthogonaltty, and osC1llatory mtegrals, Princeton Math. Ser., vol. 43, Princeton Univ. Press, Princeton, NJ, 1993

16. Robert S. Strichartz, Self-similarity on mlpotent Lie grcrops, Geometric Analysis Pluladelphia, PA, 1991), Contemp. Math., vol. 140, Amer. Math. Soc., Providence, Rl, 1992 pp.123 157.

17. E. Tchoundja, Cerleson measures for the generalized Bergman spaces VIa aT 1 -type theorem, Ark. Mat. 46 (2008), no. 2, 377-406.

18. J. T. Tyson, Globel conformal Assouad dimenston in the Hetsenberg group, Con! nn. Geom. Dyn. 12 (2008),32-57.

19. J. von Neuma.nn, Eine Spektraltheorie fUr allgemeine Operatoren emu unttaren Rcrumu, Math. Nachr. 4 (1951), 258-281 (German).

DIPARTIMENTO DI MATEMATICA, UNIVERSITA Dl BOLOGNA, 40127 BOLOGNA, ITALY E-meil eddress: arcozz1IDdm.un1bo.1t

DEPARTMENT OF MATHEMATICS, WASHINGTON UNIVERSITY, ST. LOUIS, MO 6313 USA E-meil eddress: rrlDmath. wustl. edu

DEPARTMENT OF MATHEMATICS & STATISTICS, McMASTER UNIVERSITY, HA.\llLTON ~

L8S 4K1, CANADA E-mail address:Saw6453CDNlDaol.com

~ntnt de Recherches Me,\h'ma.tlquea CRM Proceedll\gII and I ea\unt No"'. Volume 111. ~OlO

The Norm of a Truncated Toeplitz Operator

Stephan Ramon Garcia and William T. Ross

ABSTRACT. We prove severa.llower bounds for the norm of a truncated Toeplitz opuator and obtam eo curious relationship between the H2 and HOC norms of functions in model spaces.

1. Introduction

In this paper, \\e continue the discussion initiated in [6) concerning the norm of a tI1lncated Toeplitz operator. In the following, let H2 denote the classical Hardy space of the op n unit disk JI]) and Ke := H2 n (8H2)J., where 8 is an inner functi n, denote one of the so-called Jordan model spaces [2,4,7). If Hoo is the set of all bounded analytiC functions on JI]), the space KS' := Hoo n Ke is norm dense m Ke see [2 p. 83) or [9, Lemma 2.3]). If Pe is the orthogonal projection from L2 := L2 8J1]), d( 211") onto Ke and cP E L2, then the operator

18 defined on Ke and is called a truncated Toeplitz operator. Various aspects f operators were studied in [3,5,6,9,10).

IT . is the norm on L2, we let

1 I A<pll := sup{IIA<p111 : I E K9', 11111 = I}

and note that this quantity is finite if and only if A<p extends to a bounded operator on Ke. When cp E Loo, the set of bounded measurable functions on 8J1]), we have the basic estimates

o ~ lIA<p1I ~ IIcplloo. However, it is known that equality can hold, in nontrivial ways, in either of the ineqnaJjties above and hence finding the norm of a truncated Toeplitz operator can be difficult. Furthermore, it turns out that there are many unbounded symbols cp E L2 which yield bounded operators A<p. Unlike the situation for classical Toeplitz operators on lP, for a given cp E L2, there many 1/J E L2 for which A<p = A,p [9, Theorem 3.1].

2 ,

2000 MaikematU:I Sub:Jed CIa'B~ficat,on, 47A05, 47B35, 47B99, Ke!J wonts and phtrasu, Toeplitz operator, model8p~, truncated Toeplitz operator, repro

duong kernel, complex symmetric operator, conjugation. First author partially supported by National Science Foundation Grant DMS·0638789,

@2010 Amerlc&n M&them&tlcal Socl .. ,)'

69

60 S. R. GARCIA AND W T ROSS

For a given symbol r.p E L2 and inner function 9, lower bounds on the quantity (1) are useful in answering the following nontrivial questions:

(1) is Acp unbounded? (2) if r.p E Loo, is Acp norm-attaining (Le., is I\Acp\ = r.p oo)?

When e is a finite Blaschke product and r.p E HOC), the paper [61 comp tel IIAcpli and gives necessary and sufficient conditions as to when A,p = r.p 'Xl' The purpose of this short note is to give a few lower bounds on A,p for general inner functions e and general r.p E L2. Along the way, we obtain a curious relationslup (Corollary 5) between the H2 and Hoo norms of functions in Ke'.

2. Lower bounds derived from Poisson's formula

For r.p E L2, let

r 1 - IzI2 \d(1 (2) (~r.p)(z) := JaD I( - zI2r.p«() 211' ' Z E lD,

be the standard Poisson extension of r.p to lD. For r.p E C aD , the continu us functions on 81lll, recall that ~r.p solves the classical Dirichlet problem with boundary data r.p. Also note that

k ( ) ._ 1 - ~9(z)

A Z.- 1- >.z ' >., z E lD,

is the reproducing kernel for Ke [91. Our first result provides a general lower bound for A'12 \ r ()\9(Z) - 9(>') \2 dz (3) ~~~ 1-18(>')12 JaD r.p Z Z - >. 211' $

In other words,

where

sup \ r r.p(z) dV>.(Z)\ $ 1 A'ED JaD

dll>.(z):= 1_1>'12

\9(Z) - 9(>') \2 dz 1 -19(>.)12 Z - >. 211'

is a family of probability measures on alD indexed by >. E lD.

PROOF. For>. E lD we have

(4) 1-19(>')12

1_1>.12 '

A,p.

•

THE NORM OF A TRUNCATED TOEPLITZ OPERATOR 61

That the measures dv.\ are indeed probability measures follows from (4). 0

Now observe that if 9(>') = 0, the argument in the supremum on the left hand side of (3) becomes the absolute value of the expression in (2). This immediately yields the following corollary:

Corollary 1. If ep e L2, then

(5) sup 1(l.l3ep)(>')1 ~ IIA<pII, .\eS 1 ({O})

where the supremum is to be regarded as 0 if 9-1 ( {O}) = 0.

Under the right circumstances, the preceding corollary can be used to prove that certain truncated 'Ibeplitz operators are norm-attaining:

Corollary 2. Let 9 be an inner function having zeros which accumulate at every pomt of alI}. If ep e C(alI}) then IIA<p11 = Ileplloo.

PROOF. Let ( e alI} be such that lep«)1 = Ileplloo. By hypothesis, there exists a sequence >'n of zeros of 9 which converge to (. By continuity, we conclude that

lIeplloo = lim 1 (l.l3ep)(>'n)I ~ IIA<p1I :5 11epll00 n~oo

whence A<p = Ileplloc. o

The preceding corollary sta.nds in contrast to the finite Blaschke product setting. Indeed, if 9 is a finite Blaschke product and ep e H oo , then it is known that ~ = ep 00 if and only if ep is the scalar multiple of the inner factor of some

function from Ks [6]Theorem 2. At the expense of wordiness, the hypothesis of Corollary 2 ca.n be considerably

weakened. A cursory examination of the proof indicates that we only need ( to be a limit point of the zeros of e, ep e Loo to be continuous on an open arc containing (, and ep ( = ep 100'

Theorem 1 yields yet another lower bound for IIA<pIl. Recall that an inner function 9 has a finite angular derivative at ( e a]l)) if e has a nontangential limit 9 ( of modulus one at ( and 9' has a finite nontangential limit e' «) at (. This is equivalent to asserting that

6 e(z) - e«)

z-(

has the nontangential limit 9' «) at (. If e has a finite angular derivative at (, then the function in (6) belongs to H2 and

lim r~l 8D

9(z) - 9(r() 21dzl --z - r( 271' 8ID

e(z) - e«) 21dzl • •

z - ( 271'

FurtheI'more, the above is equal to

lim 1-1~(r()12 = le'(C)1 > O. r~l 1- r2

See [1,8] for further details on angular derivatives. Theorem 1 along with the preceding discussion and Fatou's lemma yield the following lower estimate for IIA<pII .

62 S. R. GARCIA AND W. T. ROSS

Corollary 3. For an inner function 9, let De be the set of < E em for whWi 9 has a finite angular derivative 9' «) at <. If c.p E LOO or if c.p E L2 with 'P ~ 0 then

1 \ r \9(Z) - 8«) \2 1dZI \ ~~~e 18'«)1 laD c.p(z) z - < 21T:5 I\Arp I· In other words,

where

d ().=_1_\9(Z)-8«)\2~ II). z. 19'«)1 z - ( 21T

is a family of probability measures on olDl indexed by < E De-

3. Lower bounds and projections

Our next several results concern lower bounds on I Arp I involving the orthogonal projection Pe: L2 ~ Ka.

Theorem 2. If 8 is an inner function and c.p E L2, then

I\ Pe(c.p) - e(O)Pa (8c.p) 1\ < I A (1 - 18(0)12)1/2 - rp'

PROOF. First observe that I\ko \I = (1 -18(0)12)1 2. Next we see that if rp E £2

and g E Ka is any unit vector, then

(1-18(0)12)1/2I\Acpl\ ~ I(Arpko,g)1 = I(Pa(c.pko),g)\ = (Pa c.p - 60 Pa e.p,9 .

Setting

yields the desired inequality. o

In light of the fact that Pe (8c.p) = 0 whenever c.p E H2, Theorem 2 leads us immediately to the following corollary:

Corollary 4. If 8 is inner and c.p E H2, then

IIPe (c.p) \I (7) (1 _18(0)12)1/2 :5 I\Acpl\.

It turns out that (7) has a rather interesting function-theoretic implication. Let us first note that for c.p E Hoo I we can expect no better inequality than

I\c.pl\ :5 1Ic.plloo

(with equality holding if and only if c.p is a. scalar multiple of an inner function). However, if c.p belongs to K~ I then a stronger inequality holds.

Corollary 5. If 6 is an inner function, then

(8) 1Ic.p1l :5 (1-16(0)12)1/21\c.p1l00

holds for all c.p E K:. If 6 is a finite Blaschke product, then equality holds if and only if c.p is a scalar multiple of an inner function from Ke.

THE NORM OF A TRUNCATED TOEPLITZ OPERATOR 63

PROOF. First observe that the inequality

lIIP11 ~ (1-le(O)12)! II IPII 00

follows from Corollary 4 and the fact that PaIP = IP whenever I{) E Ka. Now suppose that e is a finite Blaschke product and assume that equality holds in the preceding for some IP E Kif. In light of (7), it follows that IIA\OII = 111{)11"". From [6, Theorem 2] we see that IP must be a scalar multiple of the inner part of a function from Ka. But since IP E Kif, then IP must be a scalar multiple of an inner function from Ka. 0

When e is a finite Blaschke product, then Ka is a finite dimensional subspace of H2 consisting of bounded functions [3,5,9]. By elementary functional analysis, there are Cl, C2 > 0 so that

CIIIIPIl ~ IIIPII"" ~ c211IPil for all IP E Ka. This prompts the following question:

Question. What are the optimal constants Cl, C2 in the above inequality?

4. Lower bounds from the decomposition of Ka

A result of Sarason [9, [Theorem 3.1]] says, for IP E L2, that

9 A\O=O·( )'IPEeH2+eH2.

It follows that the most general truncated Toeplitz operator on Ka is of the form A",+x where t/J, X E Ka· We can refine this observation a bit further and provide an ther canonical decomposition for the symbol of a truncated Toeplitz operator.

Lemma 1. Each bounded truncated Toeplitz operator on Ka is generated by a symb 1 of the f071n

10

here t/J, X E Ka.

Before getting to the proof, we should remind the reader of a technical detail. It follows from the identity Ka = H2 n ezH2 (see [2, p. 82]) that

C: Ka -+ Ka, Cf := zfe,

is an isometric, conjugate-linear, involution. In fact, when AII' is a bounded operator we have the identity CAIpC = A; [9, Lemma 2.1].

PROOF OF LEMMA LIfT is a bounded truncated Toeplitz operator on Ka, then there exists some IP E L2 such that T = AII" We claim that this <p can be chosen to have the special form (10). First let us write <p = f + zg where I, 9 E H2. Using the orthogonal decomposition H2 = Ka EB 8H2, it follows that <p may be further decomposed as

<p = (/1 + eh) + Z(g1 + 8g2 )

where 1t,91 E Ka and h,g2 E ~. By (9), the symbols 812 and 8(Zg2) yield the zero tnmcated Toeplitz operator on Ka. Therefore we may assume that

<p = 1 + zg

for some I,g E Ka. Since Cg = gz8, we have zg = (Cg)8 and hence (10) holds with t/J = I and X = Cg. 0

64 S. R. GARCIA AND W. T. ROSS

Corollary 6. Let 8 be an inner function. If ""1. ""2 E Ke and 'P = ""1 + ¢i8 then

PROOF. If 'P = ""1 + ",,;e, then, since ""1,""2 E Ke and ""28 E zH2, we have

Pe('P) - 8(0)Pe(8'P) = ""1 - 8(0)""2' The result now follows from Theorem 2.

References

o

1. J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform, Math. Surveys Monogr vol. 125, Amer. Math. Soc., Providence, Rl, 2006.

2. J. A. Cima and W. T. Ross, The backward shIft on the Hardy space, l\1ath. Surveys Monogr vol. 79, Amer. Math. Soc., Providence, Rl, 2000.

3. J. A. Cima, W. T. Ross, and W. R. Wogen, Truncated Toephtz operators onfimte dsmenswna! spaces, Oper. Matrices 2 (2008), no. 3, 357-369.

4. R. G. Douglas, H. S. Shapiro, and A. L. Shields, CyclIc vectors and m anant subspace.9 Jmthe backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970 ,no. 1,37-76

5. S. R. Garcia, Conjugation and Clark operators, Recent Advances in Operator-Related Funotion Theory, Contemp. Math., vol. 393, Amer. Math. Soc., PI'OVldence, Rl, 2006, pp. 67 111.

6. S. R. Garcia and W. T. Ross, A nonlinear e:rlremal problem on the Hanly space, Comp t.

Methods Funct. Theory 9 (2009), no. 2, 485-524. 7. N. K. Nikol'skil, 1reatise on the shift operator, Grundlehren Math. WJSS. vol 273, Spnn,,<>er

Berlin, 1986. 8. D. Sarason, Sub-Hardy Hilbert spaces in the unit dlSk, Umv. Arkansas Lecture N tes Math.

Sci., vol. 10, Wiley, New York, 1994. 9. ___ , Algebraic properties of truncated Toeplltz operotors, Oper. Matrice; 1 2007 DO. 1,

491-526. 10. ___ , Unbounded Toeplitz operators, Integral Equatlons Operator Theory 61 2008, no 2,

281-298.

DEPARTMENT OF MATHEMATICS, POMONA COLLEGE, CLAREMONT, CA 91711, USA

E·mail address: Stephan.GarcialDpomona.edu URL: http://pages.pomona.edu/-sg064747

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, UNIVERSITY OF RICHMOND, RICHMOND, VA 23173, USA

E·mail address: wrosslDrichmond. edu URL: http://facultystaff.richmond.edu/-vross

Centre de Recherches Mathtlmatlque. CRM Proceedings and Lecture Not ... Volume 111. :1010

Approximation in Weighted Hardy Spaces for the Unit Disc

Andre Boivin and Changzhong Zhu

ABSTRACT. In this paper we study polynomial and rational approximation in the weighted Hardy spaces for the unit disc with the weight function satisfying Muckenhoupt's (Aq) condition.

1. Introduction

In [4], some basic properties of the weighted Hardy spaces for the unit disc D with the weight function satisfying Muckenhoupt's (Aq) condition were obtained, including series expansions of functions in these spaces with respect to the systems

{ 2m l-akz))-l}, with ak ED, k = 1,2 ...• In this paper, we continue our study of approximation properties in these spaces. In particular, we obtain some results on the rate of convergence of approximation by polynomial and rational functions. Let us first recall some definitions and known properties.

Assume that w is a nonnegative (with 0 < w < 00 a.e.), 211" periodic measurable fun tion defined on (-00,00). For 1 < q < 00, we say w satisfies Muckenhoupt's Aq condition or w E (Aq) (we also call w an (Aq) weight), if there is a constant

C such that for every interval I with iIi $ 211",

1

I w(O) dO

r 1 w(O)-l/(q-l) dO iIi r

q-l $ C,

where I denotes the length of I. We say w E (A 1) if

1 iIi r w(O) dO $ CIIwll[,

for every interval I with iIi $ 211", where IIwllr denotes the essential infimum of w over I.

Aq) weights were introduced in [12]. In the general definition, w is not necessarily 211" periodic and I is not restricted by iIi $ 211" (see, [8, Chapter VIi 12]). But in [4] and in the current paper, as in [12, Theorem 10] and [9, Theorem 1], w is additionally assumed to be 211" periodic since the weighted Hardy spaces we consider are over the unit disc and integration takes place over the unit circle. For

2000 Mathema.t1.Cll SubJect CIQ.llstfication.. 30D55. Key words and phrases. weighted Hardy spaces, Muckenhoupt condition.

supported by NSERC (Canada). This is the final fOlln of the paper.

@2010 American Mathematical Soclet)

65

66 A. BOIVIN AND C. ZHU

27r-periodic weights, as shown in [4], the imposition or not of the condition I $ 2Jr does not change the class of (Aq) weights (the value of the constant C appeanng in the above definition may change).

Some well-known properties of (Aq) weights include: (i) if w(O) E (Aq WIth 1 < q < 00, then w(O), [w(O)]-l/(q-l) and logw(O) are integrable on l-1r,7r' (ii) w E (Aq) if and only if w1-q' E (Aq') where 1 < q < 00 and 1 q + 1 if = 1 (iii) if wE (Aq) and qo > q, then w E (Aqo), and (iv) if w E Aq) with 1 < q < 00

then w E (Aql) for some ql with 1 < ql < q. Given w E Aq for some q wit 1 < q < 00, we denote by qw the critical exponent for w, that is, the infimum of all r's such that w E (AT). We have qw ~ 1, and w E (Ar) for all T > q

Example 1.1. Let 1 < q < 00, -1 < 8 < q -1, and

(1.1) w(O) = lei6 - ei"'IB (Le., 11 + t 8, t = 1 .

By [16, p. 236], we have w(O) E (Aq).

For w(O) E (Aq), 1 $ q < 00 and 0 < P < 00, the weighted Hardy space JIP D for the unit disc D = {z E C : Izl < I} (see [7]) is the ollection of functi DB f z which are holomorphic in D and satisfy

Ilfll~l:,(D) := sup 21 1'" If(rei9 ) Pw 0) dO < 00. r<1 7r _".

The classical Hardy space HP(D) is obtained by taking w == 1. The space £P T) is the collection of measurable functions f(t) on T = {t E C: t = I} which satISfy

1 1'" IIfllil:,(T) := 27r _,,.If(ei6 ) Pw(O) dO < +00.

For 1 $ P < 00, H~(D) and ~(T) are Banach spaces. From now on m this paper, we assume that w is an (Aq) weight for some q Wlth 1 < q < 00 and Wlth

critical exponent qw (for simplicity of writing, in lemmas and theorems involving u., we will not repeat this assumption), and in most cases, we assume that 9w 1, and by the properties of (Aq) weights mentioned a.bove, we have w E AP •

By [4] and [12, Theorem 10], we have

Lemma 1.2. Assume that qw < P < 00, then H~(D) C HPo D and L~ T C

VO(T), for some Po with 1 < Po < p, that is for any fez) E Hr;,(D),

(1.2) IIfIlHPo(D) ::; G'IIJ1 H",(D),

where G' is a positive constant independent of f.

Lemma 1.3. Assume that qw < p < 00. If fez) E H~(D), then fez) has nontangentiallimits f(t) a.e. on T (J(t) is called the boundary funcnon of fez , and f(t) belongs to L~(T) and satisfies

(1.3) lim f'" If(rei9

) - f(ei9 )IPw(0) dO = 0, ,. .... 1 -or

(1.4) r~~ L"'".If(rei6 )IPw(0) dO = L:l f(ei6 )IPw (0) dO,

and

(1.5)

APPROXIMATION IN WEIGHTED HARDY SPACES FOR THE UNIT DISC 67

where Cp is a constant depending only on p.

Lemma 1.4. Assume that qw < p < 00. If f(z) e m(D), then

(1.6) 1

27T"i f(t) dt = f(z), % E Dj '=

Itl=lt-z 0, zeC\D.

Lemma 1.5. Assume that qw < p < 00. Let lip + lip' = 1. Then, for every

bounded lmear funchonalZ E (H~(D)r, there is a function ifl(z) E ~l_p/(D) such that

for / z) E m(D).

2. Smirnov's theorem and examples

It is known (see {10, Chapter IV]) that if f(z) E HP(D) and its boundary function / t) belongs to lJ'I (T) for some PI > p, then f(z) E HPI (Smirnov's theorem). A simIlar theorem also holds for the case with (Aq) weight.

Theorem 2.1. Assume that w e (Aq) for some q with 1 < q < 00. Moreover assume that 0 p. If f(z) E HP(D) and if its boundary

n / t) E lJ' T), then f(z) E H~l (D).

Before glving the proof, let us recall that for f(t) E LI(T), the HardyLittlewood maxlmal operator is defined by

"8 1 M /(el ):= sup 0<4><" 24>

8+4> \f(eiB

)\ ds. 8-4>

By [2, p. 113], the POIsson integral

satisfies

2.1

J(z) = j(rei8):= 1 "Pr(O - t)f(eit ) dt, 211" _"

O:::;r<l,

where P r 4» is the Poisson kernel:

1- r2

Pr (4)) = 1-2rcos4>+r2 '

\z\ = r < 1

H w E (Aq) for some q with 1 < q < 00, and 1 < p < 00, by [12], the operator M / is bounded from Vw(T) into itself, that is, there is a constant Op such that for every / t) E Vw(T),

" " 1M /(ei8 )IPw(O) dO:::; Op If(ei8 )IPw(O) dO.

We are now ready to prove Theorem 2.1.

PROOF. H /(z) e HP(D), by [6, Theorem 2.7], and multiplication by plq, we have


Exponentiating and using the arithmetic-geometric mean inequality (6, p. 291, we have

If(rei6)IP/q ~ exp{-.!... f1t Pr (9 - t) loglf (eit ) IP/q dt} 27r _'"

~ -.!... f'" Pr(9 - t)lf(eit)IP/q dt. 27r _'"

But, since If(ei6 )1 E Llu: (-7r, 7r), we have If(ei6 )IP/q E L'!!t/P(T), and since qpl P> q > 1 and W E (Aq), by property (iii) of the (Aq) weight mentioned in the introduction, it follows that W E (Aqpt/P). Thus, by Lemma 1.2, If(~6) P q E Ll T). Hence, by (2.1), we have

suplf(rei6 )IP/q ~ sup -.!... j'" Pr(9 - t)lf(eit)IP/q dt ~ M( f(ei6) P q),

r<1 r<l 27r _'"

where M is the Hardy-Littlewood maximal operator. Thus, for,. < 1,

and

If(rei6 )1 ~ [M(lf(ei6 )IP/q)lq/p,

J:",lf(rei6 )IP1 W (9) d9 ~ L:I[M(lf(ei6 )IP/q)lq

PIP1

w(9) d9

= L:IM(lf(ei6 )IP/q)IWl pw(O) dO

~ c L: (If (ei6 )IP/qtPl. Pw(O)d9

= c J:,..If(ei6) IP1

w(O) dO < 00,

here in the last 2 steps, we use the facts that the operator M is bounded from L,!!t/P(T) into itself, and f(ei6 ) E .v,:; (T). Hence f(z) E H!:,' (D). 0

Note. Another proof can be obtained using [13, Theorem A4.4.51 and an important result (see [4, Lemma 2.31 or [7, p. 61): If f(z) E H~(D) then f(z)Wp(z) E HP(D) where

( 1 j'" e

it +z ) Wp(z) = exp -2 -'-t -log wet) dt ,

p7r _,. el - z zED.

As an application of Theorem 2.1, let us give an example of a function belonging to H~(D):

Example 2.2. Let w(9) = lei6 _ei"'I-1/ 2 • We know (see Example 1.1), w(8) E (Aq) for q with 1 < q < 00. Fix 4>0 with 0 < 4>0 < 7r, and define

fez) == (z - ei("'-</>o)) -1/2, zED.

It is known that fez) E HP(D) for 0 < p < 2 (see [6, p. 13, Exercise 11)\ hence fez) E H 1(D). Meanwhile, its boundary function f(e i6 ) E L~' (T) for 1 < PI < 2

lAs noted in {6], 9(Z) = (1- z) 1 is in HP(D) for every p < 1. From this it follows that 9q(Z) 0:: (1 - z)-1/'I is in HP(D) for every p < q. Thus, with a. change of variable, we have I(z) e HP(D) for 0 < p < 2.


because 11' 1 1

-11' lei9 - ei(1r-q,o») 11'1/2 . lei9 -'eh' 11/2 de < 00,

(here we note that the singularity of the first factor after the integral sign is at 11" - <Po with power pd2 < I, and the singularity of the second factor, i.e., w(e), is at 11" with power 1/2). Thus, by Theorem 2.1, I(z) E Hf:/ (D) for 1 < PI < 2.

Given a. function I(z) and an angle <P, we will use 10 Rq,(z) to denote the function obtained from I(z) by rotating z by <p, that is 10 Rq,(z) = I(zeiq,).

The following examples show that if I(z) E Hf:,(D) then 10 Rq,(z) mayor may not belong to H:;'(D).

Example 2.3. Let I be given as in Example 2.2, and consider the function

g(z) = I 0 R_q,o(z) = (ze-iq,o - ei(1r-q,o»)-1/2, ZED,

The arguments presented in Example 2.2 show that g(z) E H"(D) for 1 1

note that the singularity of the function after the integral sign is at 11" with power P 2 + 1 2> 1 when P > 1), the boundary function g(ei9

) rt ~(T) for p > 1, so by Lemma. 1.3, it follows that g(z) ~ H:;'(D) for 1 <Po but <P - <Po < 11"), as in Example 2.2, we have h z E m(D) for 1 < p < 2.

Example 2.4. Let wee) be given as in Example 2.2, and {<Pn} (n = 1,2, ... ) be a. given sequence with the properties that Q < <Pn < 11", <Pi #- <Pi for i #- j, and <Pn --t 0 as n --t 00. We will construct a function h1(Z) which is in Hf:,(D) but hi 0 R_q,,, (z) ~ H:;'(D), 1 < p < 2, for all n = 1,2, .. " Let

In(z) = (z - ei(1r-q,n»)-1/2 (n == 1,2, ... ) zED.

By Example 2.2, In(z) E H:;'(D) for 1 " - ei(1r-q,n») -1/2 (n = 1,2, ... ) zED

are in H"{D) but not in Hf:,(D) for 1 (z)= 00 lkoR_q,n(z) =lnoR-q,n(z),+ IkOK ... q,,,(z).

Thus, for 1 ,,(z) ~ Hf:,(D) since In 0 R-q,n(z) rt Hf:,(D) and Ik 0 R_4> .. (z) E Hf:,(D) for all k #- n Example 2.3).


3. Approximation in H::'(D)

Recall that e. system of functions is called complete in HI:,(D) if the closed linear span of elements of the system is the space HI:,(D)j otherwise, it is called incomplete.

3.1. Approximation by polynomials. As in the classical case, we have:

Lemma 3.1. Assume that qw 0, for T < 1 sufficiently close to 1, we have

Ilf(z) - f(rz)llm;,(D) < e.

Since the Taylor series of f(rz) converges uniformly for z ~ 1, it also converges in the topology of HI:, (D). Choosing sufficiently many terms of the series, we get a polynomial P(z) which satisfies IIf(z) - P(z)lIm;,(D) < 2e, 0

Definition. Assume that qw 0, let

(3/,w(o) = sup IIf 0 R4>l - f 0 Rcf>2 L 71' 14>1-4>21<6

Now assume that there exists a 00> 0 with (3j,w(oo) < 00 and define

(1 J1I' )1 P wh,w(8) = . sup sup 27r Ih(ei [lI+(j+l)4>J) - h ei 9+ 4> Pw () d() •

3=0,1,2, ... 14>1:56 -11'

We call Wh,w the generalized modulus of continuity of h. Note that 61 ~ ~ implies Wh,w(01) ~ Wh,w(02)'

Noting that w/,w(o) ~ (3/,w(o), and when 0 ~ 00, (3j,w 0 ~ (3f,w 00 < 00, we have, for 0 ~ 0o, w/,w(o) < 00.

lt may be expected, like in the case w = 1, that wf,w(o ~ 0 as 0 -+ 0, but unfortunately, in general, this is not guaranteed, Indeed, as shown by the following example due to G. Sinnamon, there exists a function J such that the functions f 0 R,p(z) are in HI:,(D) for all 'I/J E JR, with norms in HI:,(D) uniformly bounded, and hence the same holds for the norms of their boundary functions in Yw T by Lemma 1.3, but such that for any 0 > 0 there exist angles ,p1 and 1/J'J. with 1'l/J1 - 'l/J21 < 0 and

1 II! 0 R,p1 -! 0 R,p2I1m:,(D) ~ 2'

and hence, by Lemma 1.3, II! 0 R,p1 - J 0 R,p2I1L~(T) ~ 1 positive constant depending only on p.

Example 3.2 (G. Sinnamon). Fix p with 1 < p < 2 and define w(9) == lel9 -11-1/ 2 . Then wE Ap as in Example 2.2. For each s ~ 1 set fs(z) = (Z-S)-1 2,

where the branch cut of the square root is [0,00) so that !s(z) is analytic in D. Define

g(s,8) = IIf. 0 R91IL~(T)' It is not hard to verify that

(1) g(1,8) < 00 for 8 i= 0 (as in Example 2.2); (2) 9(1,0) = 00 (as in Example 2.3); (3) 9 is continuous on [1,2] x [-7r,7r] except at (1,0);


(4) g(s,O)~ooass~l+; (5) g(s,O) S g(l, 0) for all s, OJ (6) g(s,O) S g(s, 0) for all s, O.

Note that Property 5 does not depend on the weight w, but just on the geometric observation that for s ~ 1 and Iz\ = 1, Is - z\ ~ 11- zl. Thus 1/8(z)1 S II1(z)1 for each z with Izl = 1 and hence g(s, 0) S g(l, 9).

To get Property 6, note that for any fixed s > 1, the maximum value of the function

occurs at 0 = O. Now let On. 4>n and On be three sequences in (0,71") that decrease to zero and

satisfy

On+l + On+l < tPn < On - 6n < On < 9n + On

for n = 1.2 •.... Notice that the intervals (On - On, On + On) are all disjoint and contain none of the points rl>n.

Let M = 4 L::'_11/n2 and choose a decreasing sequence Sn, with each Sn > 1, such that

Define

n=1 g(Sn,O) zED.

For each n, 0 .. < On SO g(8n,0) ~ Mn2g(1,On). Thus,

Since H!:,(D) is a Banach space this shows that J(z) E H~(D). Now let 1/1 be an arbitrary angle.

n=1 g(sn,O) m;,(D) n=1 g(sn,O)

The inequality IOn + 1/1\ < On means that -1/1 is in the interval (On - an, On + On) and 80 it can hold for at most one n. For such an n, we use the estimate, g(sn, 9n +1/1) S 9 Bn,O) and for the other values of n we estimate as above to see that the sum is bounded by 1 + L::'=11/(Mn2) = ~. This shows that for any 1/1, J oR.p(z) E H~(D, and f 0 R,pllm:,(D) S i·

Now we show that for any 6 > 0 there exist angles 1/11 and 1/12 such that 11/11 -1/12 < 6 and


Given 0 > 0 choose N so large that ION - 4>NI < o. Set 1/;1 = -ON and 1/;2 = -tPN

IIf 0 R,p1 - f 0 R,p,IIH/:,(D)

> IlfBN 0 R9N 0 R-9NIIH1:,(D) _ L I fBn 0 R9n 0 R-9N H D

- g(SN,O) n#N g(Sn,O)

_ f: fBn 0 ~n 0 R-4J 1P D

n=1 9 Sn,O

The first term is equal to 1 and, arguing as above, each of the two sums is at most ~ so the result is at least ~ as required.

Example 3.3. Assume that qw < p < 00 and w is bounded say, w 0 < K where K is a positive constant), and f(t) E V(T). Clearly, f(t E V T) a particular case is f(t) E Ll:,(T) with w = 1). In this case, we can prove that wf,w(o) --+ 0 as 0 --+ 0: Given an e > 0, there is a polynomial P z such that

\If - P\lLP(T) < e/(3K),

hence,

IIf - pIIL~(T) < c/3.

For any 4>1 and 4>2, we have

and similarly,

Ilf 0 R2 - P 0 R2I1L~(T) < c 3.

Meanwhile, by the uniform continuity of p( z) on T, when 4>1 - 4>2 is sufficiently small, we have

lip 0 Rl - P 0 R21IL~(T) < e 3.

Thus, combining the above 3 inequalities, by Minkowski's inequality, when 14>1 - 4>21 sufficiently small, we have

hence

IIf 0 Rl - f 0 R2I1L~(T) < c,

limsup \If 0 Rl - f 0 R2I1L~(T) ~ c, 1<1>1-<1>21-+0

and the required result follows as c can be arbitrarily small.

This is a very strong condition on the weight w, but it guarantees that wf,w(o) --+ 0 as 0 --+ 0 for all f E £1>(T). Simple conditions involving not only w but also f can easily be given to get wf.w(o) --+ 0 as 0 --+ O. It would be interesting to obtain a set of necessary and sufficient conditions (on w and f) for w/,w(o) --+ 0 as 0 -t 0 to hold, as it would have implications on our next two theorems. See below.

Lemma 3.4. For any positive integer k,

(3.1)


PROOF. By definition,

wh,w(k6)

= sup sup 1 "'lh(ei[9+(Hl)k4>1) _ h(ei [9+;k4>lW'w(O) dO J=0,1,2 ••.. t4>tSo 271' -".

-- sup J=0,1,2, .•

lip

lip

1 lip

sup sup --J=0,1,2, ••• 4> So

which, by Minkowski's inequality,

Ie-I 1 ". lip $ sup sup L \h(ei[9+(;k+l+l)4>l) - h(ei [9+(jk+I)4>lWw(O) dO

J-O,1,2 .... 4> So 1=0 271' -".

Ie-I

$ L sup sup 4> S6 1 lh(ei[9+(jk+I+1)4>1) - h(ei [9+(;k+I)4>lWw(O) dO 1-0 J-O,1,2.... 271' -".

". lip

$ k . w/a..w( 6).

Using a classical method similar to that in [14], we obtain

o

Theorem 3.5. Assume that qw 0 with (3h.w(60 ) < 00. Then for any positive integer m, there is a polynomial

Pm z oj order $ m such that

3.2 1

lIh(t) - Pm(t) II Ll:, (T) $ C(h)· Wh.w m '

whe,-e C h) is a constant depending on h but not m.

PROOF. Note that h(t) E Ll(T), as in [14, Chapter II!], define

where

Then

and

"./2 . t h( ei(9+2t» sm ~

-"./2 m sm t

sinmt

msint

4

dt.

"./2 [h(ei9 ) _ h(ei(9+2t»]

-"./2

4

"./2 \h(ei (9+2t» _ h(ei9 )\

-"./2

Using Holder's inequality, we have

"./2 Ih(ei(9+2t» _ h(ei9 )\p

-11' 12

dt,

sinmt

msint

sinmt

msint

sinmt

msint

4

dt,

4

dt.

4p lip

dt


where Cl == 1rl/'P' with lip + liP' = 1. Thus, using Fubini's theorem, we obtain

i:1h(eiB) - 1mUJ)l'Pw(9) d9

:5 (2Cl)'P J"/2 rf1f Ih(ei (9+2t)) _ h(eiB )I'Pw(9) dol (Sin~t)4p dt Zm -,,/2 L -1f J mSllt

5; (2ZC1)'P f"/2[Wh.,w(12t IW (Sin ~tt)4p dt. m -1f/2 mSll

By (3.1), wh,w(ko) :5 kwh.,w(o), it follows that

IIh(eiB) - 1m(9) Ii Ll:,(-",1f) :5 ~~~ (10

1f/2

[Wh.,w(t))p(::s::) 4py p.

Noting that, for t > 0,

Wh,w(t) = Wh.,w (:) :5 Wh.,w ([mt~+ 1) :5 Umt} + 1 Wh.,w(~)

:-:; (mt + 1 Wh.,w (~) (note that, here, and in the following, as usual, for a real number s, we use [s1 to denote the greatest integer not over s), we have

a ~ 0, b ~ 0, 1 < P < 00.

But we have the estimates (see [14, pp. 84-85]):

101f

/2 (sin mt )4P

C4 -- dt<-

o msint - m'

and

101f

/2

(sinmt )4P Cs tP -- dt<--

o msint - mP+1 '

where Ci (i ;: 1, ... ,5) are constants independent of m. Thus, we obtain

IIh(eI9) - 1m (0) II Ll:,(-1f,1f) :-:; C(h) . Wh.,w (~).

Noting that, by [14, Chapter III, p. 91}, Im(O) can be re-written as a polynomial Pm(z) of z = ei9 with order m - 1. The lemma is proved. 0

Remark 3.6. If Wh.,tu(o) -+ ° as 0 -+ 0, this is a Jackson-type theorem.

We will need the following fact in the next section:


Lemma 3.7. Let qw 0,

(3.3)

Then, for r > 1,

(3.4) crm + l

I\PmI\Ll:.(It\=r) ~ r -1 IIPmIlLl:.(T),

where c is a constant independent of m.

PROOF. Consider the function

Pm(Z) (3.5) f(z):= m+1' Z =F O.

Z

Noting that 1(00) = 0, by the Cauchy formula, for any tEe with It I = r > 1,

I(t) = - 1. f(e) de. 2m \~\=l e - t

Using Holder's inequality with lip + lip' = 1, we have

1 t) < 1 I/(e)1 Idel < 1 If(e)1 Idel - 21l' \~ =1 It - el - 21l' \~I=l It I - lei

< 1 1 .". \f(ei6)IPw(O) - r - 1 21l' _.".

lip 1 .". [w(OW-pl

dO 21l' _.".

IIp'

Cl = r _ 1 II fll Ll:. (T) ,

where Cl = ( 1 21l') J":.".[w(O)P_pl dO//p/

• Thus, by (3.5), for It I = r > 1,

since f Ll'o T = IPmIlLl:.(T)' Hence, by (3.3),

C1C2rm+l I\Pm\lLl:.(\t\=r) ~ 1 \I Pm \I Ll:. (T)' r-

where C2 = ( 1/21l') C.". w(O) dO//P• Letting C = C1C2, (3.4) follows. o

3.2. Approximation by rational functions. For a sequence {ak} C D, consider the system of rational functions

1 1 (3.6) ek(z) = 2 .' 1 ' k = 1,2, .. \ •

1l'1 - akZ

In [4] we studied the system {ek(z)} under the assumption that the sequence {ak} satisfies the Blaw::hke condition

00

~)l-lak\) < +00. k=l

In particular, it was shown that, under this condition, the system {ek(z)} is incomplete in HI:,(D), for qw < P < 00. See [4, Lemma 3.3]. We then proceeded to study the subspace generated by the system {ek(z)}.

76 A. BOIVlN AND C. ZHU

In the current paper, we consider the case when the sequence {a,.} doe! not satisfy the Blaschke condition.

Lemma 3.S. Assume that qw < p < 00. If 00

(3.7) ~)l-\a,.1) = +00, ,.=1

then the above system {e,.(z)}(k = 1,2, ... ) is complete ~n H{;, D .

PROOF. By the Hahn Banach theorem, and Lemma 1.5, we need nly to prm-e that if

k = 1,2 ...

where ~(z) E H~l_P' (D) with l/p + l/p' = 1, then ~ e,8 = 0 B.e. in -'11"," But (3.8) is equivalent to

2.. r ~(t) dt = 0 211'i lltl=l t - a,. ,

that is, by Lemma 1.4, ~(a,.) = 0 (k = 1,2, ... ). We note that, by Lemma 1.2, ~(z) E HB for some 1 < s < p'. Thus, by (3.7) and Corollary f (6, Theorem 23 we have ~(z) == O. The lemma is proved. 0

It follows that, under the assumptions of the ab ve lemma, v.e can use linear combinations of the system {e,. (z)} (k = 1,2, ... ) to approximate any functi n in H{;,(D). Assume that {a,.} contains the point zero. Without loss of generality by re-indexing, we assume that ao = 0 and all a,. ::f 0 (k = 1,2, .•.. Thus, for a fixed positive integer n, a linear combination of the system becomes

n

(3.9) ~ C,.

Tn(Z) = Co + ~ 1 . "'=1 - a/tz

The poles of Tn(Z) are b,. = 1/0.", with \b,.1 > 1 (k = 1,2, ... , n . Denote by 'R.n the collection of all rational functions rn(Z) of the above form, and, for h Z E H~ D , denote the best approximation value of h by r n in R... by

By Lemma 3.8, we have En(h) -+ 0 as n -+ 00. An interesting question is how to estimate the speed of En(h) -+ O. Similar problems were studied for uniform approximation on T or D (see (lD. and for the approximation in HP (see (15]).

Theorem 3.9. Assume that (i) qw 0 with fJh,w(OO) < 00; (ii) {a",} satisfies (3.7) and la",1 ::; p (k = 1,2, ... ) with 0 2, "'=1

and

(3.11) (2) (l-p)ut3 n - :5 wh,w(l)

e


(here, assume that wh,,.,(I) < 00), then there exists a rational function rn(z) E'R.n such that

(3.12)

where C is a positive constant depending on h but not n, and wh,,.,(c5) is the generalized modulus of continuity of h. Hence, we have

(3.13)

PROOF. Choose a positive integer m = [s .. /2]. By (3.10), m ~ 1, and by 3.5, there is a polynomial Pm (z) of degree $ m such that

(3.14) 1

I\h(t) - Pm(t)I\Ll:,(T) $ C1 • Wh,,., , m

where Cl is a const.ant depending on h but independent of m. Noting that wh,,.,(I/m) $ wh,,.,(I) (since l/m $ 1), by (3.14), we have

3.15 Pm LP (T) $lIhI\Ll:,(T) + IIh - Pm \ILl:, (T) $l\hI\Ll:,(T) + C1 • Wh,w(l) = C2,

where C2 is a constant depending on h but not m and n. Assume that a rational function Tn(Z) E 'R.n interpolates Pm(Z) at ak (k =

1,2, .•. ,n+ 1). By [17, Chapter VIII, Theorem 2], the error Pm(z) -Tn(Z) has the following integral representation: for \z\ $ 1,

3.16

where r > 1 and r # \bk\ (k = 1,2, ... ). Using (3.16), now we estimate the error \Pm(Z) - Tn(Z)\ in \z\ $ 1:

Itl=r

1 =-

211" Itl=r

Clearly, for Izi $ 1,

n . IT Z - ak

Z - bk k=l -

z- bk k=l

n

IT ak-Z = Iz - Bn+1\ 1 __

k=l Uk z

\Pm(t)\\dt\ \t - z\

\Pm(t)\\dt\. It - z\

= \z - a"+1\IB .. (z) 1 S (Iz\ + lan+lDIBn(z)1 $ 2.

And, by {17, Chapter IX, Section 2, Lemma, p, 229], we have for It I ::: r > 1,


So, for Izi ~ 1, by Holder's inequality,

< 2 rrn Ibkl + r I! (3.17) IPm(z) - rn(z)1 - (r -1)2 . 1o~1 1 + Ibklr . 271' t =r Pm(t) dt

< 2C1r rrn Ibkl + r - (r - 1)2 • 10=1 1 + Ibklr ·1 Pm\lL~( t =7'),

where Cl = (1/271')J~n[w(lI)]I-P' dll)I/P' with 1/p+1/P' = 1.

By Lemma 3.7, and taking r = 2 (note that if Ibk = 2 for some k, we can choose r = 2 + e with a sufficiently small positive number e), we have for z ~ 1,

m rrn Ibkl +2 (3.18) IPm(Z) - rn(z)1 ~ C22· 1 + 21b I' Pm LP T).

10=1 10

where C2 is a constant independent of m and n. Therefore, by (3.18 and 3.15, we have

(3.19) m rrn Iblol +2 I\Pm - rnI\Ll:,(T) ~ C3 • 2 1 + 2 b '

10=1 k

where C3 is a constant depending on h but not m and n. And by 3.19 and 3.14, noting that 2m = 2[sn/2] ~ Sn, we obtain:

( 1) S IT" bk + 2 (3.20) I\h - rnI\Ll:,(T) ~ C 1 . Wh,w [sn/2] + C 3 • 2 n k=1 1 + 2 bk .

Let us estimate the product in the right-hand side of the above inequality. Since

Ibkl +2 0< 1+21bkl < 2,

we have2

Ibkl + 2 (Ibkl + 2) (bk -1) 1 + 21 bkl < exp 1 + 21bkl -1 = exp 1 + 2 bk •

Since Ibkl > 1, noting (3.10), we have

rrn Ibkl + 2 < (~Ibkl- 1 ) (~ bk -1) 10=1 1 + 21bkl - exp - ~ 1 + 21bkl < exp - ~ 3 bioi

= exp(-! t(l- --l-)) = exp(-! t(l- lak \)) = e-sn

• 3 k~l I 101 3 k=1

Thus, by (3.20), it follows that

(3.21) I\h - TnI\Ll:,(T) ~ C1Wh,w ([sn1/ 2]) + C3 (~r"·

Since lakl ~ p (k = 1,2, ... ), we have

1 n 1 n 1 871 = 3 L(l-lak\) ~ 3 L(l - p) = "3n (l ~ p).

10=1 k~l


Hence

8 n 1 ( ) > -1-p n 2 - 6 .

Since Wh,..,(O) is nondecreasing as 0 increasing, we have by the inequality wh,w(ao) :5 (a + l)wh,..,(o) for a > 0,

where

1 1 :5 O •. Wh,.., - ,

n

1 O. = [~(1- p)]'

Meanwhile, noting that 2 e < 1, we have

Let

2 8ft 2 (1-p)n/3

- < --e e

2 (1-p)/3

).= -e

,

•

then clearly we have 0 < ). < 1. Thus, by (3.21), it follows that

IIh - rnIlLl:,(T) :5 Os Wh,.., .!. + .).n , n

3.22

where 0 5 is a positive constant depending only on f and p. But

1 1 wh,w(l) = Wh,.., n· - :5 nwh,w - .

n n

Hence, by 3.11),

Wh,..,

Thus, 3.22) implies

3.23)

- -n n e

1 IIf - rnIlLl:,(T) :5 0 . Wh,.., - , n

and we have (3.12) since h - rn E H~(D) and, by Lemma 1.3, the two norms of h - rn H1:.(D) and IIh - rnIlLl:,(T) are equivalent. The proof is complete. 0

Remark 3.10. If Wh,..,(O) -+ 0 as 0 -+ 0, (3.13) gives an estimate of the speed of En{h) -+ O.

Acknowledgement. We are indebted to the referee for his/her careful reading and valuable suggestions and corrections which greatly improved this paper, and to G. Sinnamon for helpful discussions and providing us in particular with Example 3.2.


References

1. J .-E. Andersson and T. GaneliuB, The degree of appro:nmat~on by ratwnal /unc:t.wM 1UIlh /i:u4 poles, Math. Z. 153 (1977), no. 2, 161 166.

2. S. Axler, P. Bourdon, and W. Ramey, Harmonu: /unct~on theory, Grad. Texts in Math., vol. 137, Springer, New York, 1992.

3. E. F. Beckenbach and R. Bellman, Inequal,bes, Ergeb. Math. Grenzgeb., vol. 30, Spnnger New York, 1983.

4. A. Boivin, P. M. Gauthier, and C. Zhu, Weighted Hardy spaces for the fond duc' appnnmlatwn properties, Complex and Harmonic Anal)'Bis, DEStech Publ., Lancaster, PA. 2007, pp 129-155.

5. J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform, Math Surveys MoDOgl' vol. 125, Amer. Math. Soc., Providence, RI, 2006.

6. P. L. Duren, Theory of HP spaces, Pure Appl. Math., vol. 38, Academic Press, New York, 1970; Dover, Mineola, NY, 2000.

7. J. Garda.-Cuerva, Weighted HP spaces, Dissertationes Math. Rozprawy Mat. 162 1979 8. J. B. Garnett, Bounded analyt,c /unctIOns, Pure Appl. Math., vol. 96, AcadeIDlc Press, New

York, 1981. 9. R. Hunt, B. Muckenhoupt, and R. Wheeden, WeIghted norm nequal bes for the conJUgate

function and Hilbert transform, Trans. Amer. Math. Soc. 176 1973,227-251. 10. P. Koosis, Introduction to Hp spaces, 2nd ed., Cambridge Tracts in Math., vol. 115 Cambndge

Univ. Press, Cambridge, 1998. 11. J. Lee and K. S. Rim, Weighted norm inequahties for plunharmonlc ctm]U9ate functaona, J

Math. Anal. Appl. 268 (2002), no. 2, 707-717. 12. B. Muckenhoupt, Weighted norm inequabtles for the Hardy maxrmal function, Trans. AmEl'

Math. Soc. 165 (1972),207-226. 13. N. K. Nikolski, Operators, /unctions, and systems: an easy read g. Vol. 1: Hardy Hanke/,

and Toeplitz, Math. Surveys Monogr., vol. 92, Amer. Math. Soc., Providence, RI, 2002, VoL 2: Model operators and systems, vol. 93.

14. W. E. Sewell, Degree of approximation by polynomIals in the comple:!: domam, AIm.. of}.1ath Stud., vol. 9, Princeton Univ. Press, Princeton, NJ, 1942.

15. X. C. Shen and Y. R. Lou, Best approXImation by rabonal functions n Hp spaces (p ~ 1 , Beijing Daxue Xuebao 1 (1979), 58-72 (Chinese).

16. A. Torchinsky, Real-variable methods in harmonIC analySIS, Pure Appl. Math., vol. 123, Academic Press, Orlando, FL, 1986.

17. J. L. Walsh, Interpolation and approximabon by rabonal funcbOfl$ m. the comples domasn, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., ProVIdence, R.I., 1965.

DEPARl'MENT OF MATHEMATICS, UNIVERSITY OF WESTERN ONTARIO, LONDON, ON N6A 5B7,

CANADA

E-mail address.A.Boivin: b01vinlDuvo.cll E-mail address, C. Zhu; czhu28lDuvo. CIl

Centre d. Recherche. Math<lmatlquea CRM Proceedings and I ecture Notes Volume Ill, 2010

Some Remarks on the Toeplitz Corona Problem

Ronald G. Douglas and Jaydeb Sarkar

ASSTRACf. In a. recent paper, Trent and Wick [23] establish a strong relation between the corona problem and the Toeplitz corona problem for a family of spaces over the ball and the polydisk. Their work is based on earlier work of Amar [3]. In this note, several of their lemmas are reinterpreted in the Ja.nguage of Hilbert modules, revealing some interesting facts and raising some questions about quasi-free Hilbert modules. Moreover, a modest generalization of their result is obtained.

1. Introduction

While isomorphic Banach algebras of continuous complex-valued functions with the supremum norm can be defined on distinct topological spaces, the results of Gelfand cf. [12]) showed that for an algebra A ~ C(X), there is a canonical ch ice of domain, the maximal space of the algebra. If the algebra A contajns the function 1, then its maximal ideal space, MA, is compact. Determining MA for a concrete algebra is not always straightforward. New points can appear, even when the riginal space X is compact, as the disk algebra, defined on the unit circle T, demonstrates. If A separates the points of X, then one can identify X as a subset

f MA with a point Xo in X corresponding to the maximal ideal of all functions in A vanishing at Xo. When X is not compact, new points must be present but there is still the question of whether the closure of X in MA is all of MA or does there exist a "corona" MA X =f 0.

The celebrated theorem of Carleson states that the algebra HOO(][J)) of bounded holomorphic functions on the unit disk ][J) has no corona. There is a corona problem for HOO(n) for every domain n in em but a positive solution exists only for the case m = 1 with n a finitely connected domain in e.

2000 MathematU:8 SubJect Clas8Jji.cahon. 46E50, 46E22, 47832, 50H05. Key warda and phratJetI. Corona problem, Toeplitz corona problem, Hilbert modules, Kernel

HilbCi t spares. This rveaTch was supported in part by a grant from the National Science Foundation. This is the final version of the paper.

©2010 Amerloan Mathematlc,,1 Society

81

82 R. G. DOUGLAS AND l. SARKAR

One can show with little difficulty that the absence of a corona for an algebra A means that for {'Pi}f=l in A, the statement that

n

(1) Li'P,(X)\2 ~ e2 > 0 for all x in X 0=1

is equivalent to

(2) the existence of functions {1JI'}~=1 in A such that n

L 'P,(x)1JI.(x) = 1 for:r: in X . • =1

The original proof of Carleson [8] for HCXl(lDl) has been simplified over the years but the original ideas remain vital and important. One attempt at an alternate approach, pioneered by Arveson [6] and Schubert [20], and extended by AglerMcCarthy [2], Amar [3], and finally Trent-Wick [23] for the ball and poly disk, involves an analogous question about Toeplitz operators. In particular, for {.p }~1 in HOO(n) for n = Bm or lDlm , one considers the Toeplitz operator T4>: H2 Q n-t

H2(n) defined T4>f = E~=l 'Pdi for f in H2(Q), where f = h fJ) ••• fJ) In and X" = X E9 ... E9 X for any space X. One considers the relation between the operator inequality

(3) for some E > 0

and statement (1). One can readily show that (3) implies that one can soh-e 2 where the functions {1JIi}~=1 are in H2(Q). We will call the existence of such functions, statement (4). The original hope was that one would be able to modify the method or the functions obtained to achieve {1/..} ~1 in Hoo Q. That 1 implies (3) follows from earlier work of Andersson-Carlsson [5] for the unit ball and of Varopoulos [24], Li [17], Lin [18], Trent [22] and Treil-Wick. [21] for the polydisk.

In the Trent - Wick paper [23] this goal was at least partially accomplished with the use of (3) to obtain a solution to (4) for the case m = 1 and for the case m > 1 if one assumes (3) for a family of weighted Hardy spaces. Their method was based on that of Amar [3].

In this note we provide a modest generalization of the result of Trent - Wick in which weighted Hardy spaces are replaced by cyclic submodules or cyclic invariant subspaces of the Hardy space and reinterpretations are given in the language of Hilbert modules for some of their other results. It is believed that this reformulation clarifies the situation and raises several interesting questions about the corona. problem and Hilbert modules. Moreover, it shows various ways the Corona Theorem could be established for the ball and polydisk algebras. However, most of our effort is directed at analyzing the proof in [23] and identifying key hypotheses.

2. Hilbert modules

A Hilbert module over the algebra A(Q), for n a bounded domain in ern, is a Hilbert space 'H. which is a unital module over A(Q) for which there exists C ~ 1 so that II'P' III'H ::; C\\'PIIA(O)II/II'H for'P in A(O) and I in 'H.. Here A(Q) is the closure in the supremum norm over 0 of all functions holomorphic in a neighborhood of the closure of Q.

SOME REMARKS ON THE TOEPLITZ CORONA PROBLEM 83

We consider Hilbert modules with more structure which better imitate the classical examples of the Hardy and Bergman spaces.

The Hilbert module 'R over A(O) is said to be quasi-free of multiplicity one if it has a canonical identification as a Hilbert space closure of A(O) such that:

(1) Evaluation at a point z in 0 has a continuous extension to 'R for which the norm is locally uniformly bounded.

(2) Multiplication by a rp in A(O) extends to a bounded operator TIP in £('R). (3) For a sequence {rpk} in A(O) which is Cauchy in 'R., rpk(Z) -+ 0 for all z in

o if and only if IIrpk II'R -+ O. We normalize the norm on 'R so that IIIII'R = 1. We are interested in establishing a connection between the corona problem for

M('R.) and the 'lbeplitz corona problem on R. Here M(R) denotes the multiplier algebra for 'R.j that is, M (= M('R.» consists of the functions 'IjJ on 0 for which t/J'R. C 'R.. Since 1 is in n, we see that M is a subspace of R and hence consists of holomorphic functions on O. Moreover, a standard argument shows that 'IjJ is bounded (cf. [10]) and hence M C HOO(O). In general, M i- HOO(O).

For .,p in M we let T", denote the analytic Toeplitz operator in £(R) defined by module multiplication by'IjJ. Given functions {rpdf:,l in M, the set is said to

1) satisfy the corona condition if L:~=ll rpi (z) 12 ~ E:2 for some E: > 0 and all z in OJ

2 have a corona solution if there exist {'ljJi}f=l in M such that L::l rpi (Z)'ljJi(Z) = 1 for z in OJ

3 satisfy the Toeplitz corona condition if L:~=1 TIP. T;, ~ E:2 I'R for some E: > 0; And

4 satisfy the 'R.-corona problem if there exist {fi}?=l in R such that L:~=1 TIP. Ii = 1 or L:~=1 rp.(z)f(z.) = 1 for z in 0 with L::lIIIiI12 ::; I/E:2.

3. Basic implications

It is easy to show that (2) :. (1), (4) :- (3) and (2) :- (4). As mentioned in the introduction, it has been shown that (1) :- (3) in case 0 is the unit ball Bm or the polydisk][»m and (1) )- (2) for n =][» is Carleson's Theorem. For a class of reproducing kernel Hilbert spaces with complete Nevanlinna- Pick. kemels one knows that (2) and (3) are equivalent [7] (cf. [4,15]). These results are

related to generalizations of the commutant lifting theorem [19]. Finally, 3 :. (4) results from the range inclusion theorem of the first author as follows cf. [11]).

Lemma 1. If {rp.}f:,l in M satisfy L::1 TIPiT;. ~ E:2 In. for some E: > 0, then there e",ist {f'}~1 ~n R such that L:~1 rpi(z)li(z) = 1 for z in n and L:~lllldl~ ::; 1 g2.

PROOF. The assumption that L:~1 T IPi T;, ~ E:2 I implies that the operator

X: 'R.n -+ 'R defined by XI = L:~=lTIP11i satisfies XX· = L::=lTIP,T;, ~ E:2In. and hence by [11] there exists Y: R -+ Rn such that XY = In. with IIYII ::; ~, Therefore, with YI = It E9 •. , E9 fn. we have E~=l rpl(Z)/i(Z) = E~l TIP"i = XYI = 1 and L:~=1111i1l~ = I!YIII2 ::; 11Y1I211III~ ::; I/E:2. Thus the result is proved. 0

To compare our results to those in [23], we need the following observations.

84 R. G. DOUGLAS AND J SARKAR

Lemma 2. Let n be the Hilbert module L~(J.L) over A(O) defined to be the closure of A(O) in L2(J.L) for some probability measure J.L on cl08 O. For J m PCp , the Hilbert modules L~(\f\2 dJ.L) and [J), the cyclic submodule oJR genemted by f, are isomorphic such that 1 -+ J.

PROOF. Note that 11'P ·1\\L2(lfI2 dl') = 11'PJIIL2(1') for 'P in A(O and the closure of this map sets up the desired isomorphism. 0

Lemma 3. If {h}i::l are functions in L~(J.L) and g(z) = L:~=l J,(z 2, then L~(gdJ.L) is isomorphic to the cyclic submodule [ft ffi··· ffi Jn1 oj L~(p. n unth 1 ~ It ffi ... ffi In.

PROOF. The same proof as before works. o In [23], 'frent - Wick prove this result and use it to replace the L~ spaces used

by Amar [3] by weighted Hardy spaces. However, before proceding we want to explore the meaning of this result from the Hilbert module point of view.

Lemma 4. For n = H2 (Il~m) (or H2 (JIlm)) the cyclic submodule oj R n generated by 'Pl ffi ... ffi 'Pn with {'Pi}i::l in A(Jam) (or A(JIlm)) is isomorph'c to a cychc submodule oj H2(JBm) (or H2(JIlm )).

PROOF. Combining Lemma 3 in [23] with the observations made in Lemmas 2 and 3 above yields the result. 0

There are several remarks and questions that arise at this point. First, does this result hold for arbitrary cyclic submodules in H2 (Jam n or H2 Jl))m n, which would require an extension of Lemma 3 in [23] to arbitrary f in .H2(Bm n or H2(JIlm)n? (This equivalence follows from the fact that a converse to Lemma 2 is valid.) It is easy to see that the lemma can be extended to an n-tuple of the form hh EB ... ffi Jnh, where the {fi}i::l are in A(O) and h is in 'R.. Thus one need only assume that the quotients {hi f; }f,j=l are in A(O) or even only equal a.e. to some continuous functions on a~.

Second, the argument works for cyclic submodules in H2(Bm ®12 or H2(Dm ® l2 as long as the generating vectors are in A(O) since Lemma 3 in [23] holds in this case also.

Note that since every cyclic submodule of H2(JIl) ® l2 is isomorphic to H2(D , the classical Hardy space has the property that all cyclic submodules for the case of infinite multiplicity already occur, up to isomorphism, in the multiplicity one case. Although less trivial to verify, the same is true for the bundle shift Hardy spaces of multiplicity one over a finitely connected domain in e [1].

Third, one can ask if there are other Hilbert modules R that possess the property that every cyclic submodule of 'R ® en or R ® l2 is isomorphic to a submodule of n? The Bergman module L~ (JIl) does not have this property since the cyclic submodule of L~(JIl) ffi L~(JIl) generated by 1 ffi z is not isomorphic to a submodule of L~(JIl). If it were, we could write the function 1 + Izl2 = If(z)12 for some J in L~(JIl) which a simple calculation using a Fourier expansion in terms of {z"zm} shows is not possible.

We now abstract some other properties of the Hardy modules over the ball and polydisk.

We say that the Hilbert module 'R over A(O) has the modulus approximation property (MAP) if for vectors {J'}~l in M ~ 'R, there is a vector k in'R such tha.t


lI{lkll~ = ~f=lll{l fJ 1\2 for {I in M. The map {lk -+ {If. e ... e {If N thus extends to

a module isomorphism of [k] en and [f1 e··· e fN] c nN. For Zo in n, let 1"0 denote the maximal ideal in A(n) of all functions that

vanish at Zoo The quasi-free Hilbert module n over A(n) of multiplicity one is said to satisfy the weak modulus approximation property (WMAP) if

(1) A nonzero vector kza in n e 1"0 • n can be written in the form kzo . 1, where kza is in M, and Tk.o has closed range acting on n. In this case n is said to have a good kernel function.

(2) Property MAP holds for J. = A.k"j, i = 1, ... , N with 0 ~ A~ ~ 1 and

~!1 A~ = 1.

4. Main result

Our main result relating properties (2) and (3) is the following one which generalizes Theorem 1 of [23].

Theorem. Let n be a WMAP quasi-free Hilbert module over A(n) of multiplicity one and {<'o1}:=1 be functions in M. Then the following are equivalent:

(a) exist functions NiH=l in Hoo (n) such that ~~=l ct'i(Z)Wi(Z) = 1 and ~ w.(z)1 ~ 1/£2 for some £ > 0 and all Z in n, and

(b) there exists £ > 0 such that for every cyclic submodule 5 ofn, ~~=l T~, T~: ~ c2 Is, where T~ = T<pls for ct' in M.

PROOF. We follow the proof in [23] making a few changes. Fix a dense set {Z.}~2 of n.

First, we define for each positive integer N, the set CN to be the convex hull of the functions {lkz.12 Il\kz.1\2}~2 and the function 1 for i = 1 with abuse of notation. Since'R being WMAP implies that it has a good kernel function, CN consists of nonnegative continuous functions on n. For a function g in the convex hull of the set { kz, 2/ k", 12}!1' the vector Al k"J II k"J 2 e ... e AN kZN I II kZN 112 is in nN. By definition there exists Ginn such that [G] ~ [AlkzJllkz, II e ... E9 ANkzN/llkzN III by extending the map {lG -+ Al{lkzJllkz, lie··· E9 ANlIkzN/llkzN II for (J in M.

Second, let {<'ol, .•• , <,On} be in M and let T denote the column operator defined from'Rn to n by T(h e··· e In) = ~~=l T<p.fi for 1 = (ft e··· E9 fn) in nn and set Ie = kerT C nn. Fix 1 in nn. Define the function

FN: eN x Ie -+ [0,00)

by

0=1 where 9 = ~:':l A~ Ik".12 IlIkz .1I2 and ~~=l A~ = 1. We are using the fact that the k%i are in M to realize kz, (I - h) in n n.

Except for the fact we are restricting the domain of F N to C N X Ie instead of CN x nn, this definition agrees with that of [23]. Again, as in [23], this function is linear in 9 for fixed h and convex in h for fixed g. (Here one uses the triangular inequality and the fact that the square function is convex.)

Third, we want to identify F N(g, h) in terms of the product of Toeplitz operators (rZg

) (rZ9 )* , where 59 is the cyclic submodule of'R generated by a. vector P in n as given in Lemma 3 such that the map P -+ (AlkzJllkz,1I E9 ... E9 ANk"N IlIkzN II

86 R. G. DOUGLAS AND J. SARKAR

extends to a module isomorphism with 9 = ..... N ).21k 12/llk 112 0 < ).2 < 1 and L....=l • z~ "', - ,- ,

E!l),~ = 1.

Note for I in'R'\ infhEIC FN(g, h) :5 1/e2I1T~/1I2 if T!g(T!9)- ~ £2Is._ Thus, if Tg (Tg)- ~ e2 Is for every cyclic submodule of 'R, we have infhEIC F N(g, h ~ 1/e2I1T~/1I2. Thus from the von Neumann min-max theorem we obtain infhEIC SUPgECN FN(g, h) = SUPgECN infhEIC F N(g, h) ~ 1/e2 T~I 2.

From the inequality T~T; ~ e2 I'R,. we know that there exists 10 in R n such that 11/011 ~ 1/e1l111 = l/e and T~/o = 1. Moreover, we can find hN in}C such that FN(g, hN) ~ (1/e2 + 1/N)IIT~/01l2 = 1/e2 + l/N for all 9 in eN. In particular, for

gi = Ikz, 12 /lIkz, 11 2, we have T!g, (T!")- ~ e2 Is." where kz.l kz 10 - hN 2 < 1/e2 + l/N.

There is one subtle point here in that 1 may not be in the range ofTg. However, if P is a vector generating the cyclic module Sg, then P is in M and Tp has closed range. To see this recall that the map

8kz1 8kzN

8P -t ),lllkz11l EEl ••• EEl ).N I\kzN I for 8 in M is an isometry. Since the functions {kz./llkz. }~l are in M by assumption, it follows that the operator Mp is bounded on M ~ R and has closed range on 'R since the operators M", • .; II "'z, II have closed range, again by assumption. Therefore, find a vector I in S; so that T4!1 = P. But if f = II EEl··· EEl In, then

fi is in [P] and hence has the form Ii = P ii for i. in R. Therefore, T4!Tp i = P or T~i = 1 which is what is needed since in the proof 10 - f is in !C.

To continue the proof we need the following lemma.

Lemma 5. If Zo is a point in nand h is a vector \n Rn., then h zo &. S Ilkzo/llkzo IIh1l2.

PROOF. Suppose h = hI EEl ... EEl hn. with {h'}~1 in A n). Then 1h.kzo = hi(zo)kzo and hence

hi (zo)llk .. oIl2 = (Th.,kzo,k:co) = (kzo,Th,k:co)

since Tk.ohi = Th,kzo' (We are using the fact the kzoha = kz ha·1 = h.kzo ·1 = hikzo ·) Therefore,

Ihi(Zo)llIk"01l2 = l(kzo,T""oh,)1 :5l1k:co1l2

\1T"'"ol "'''0 h.\I,

or,

Finally, n

IIh(zo)II~" = Llhi (zo)12 :5 liT ""0 111""0 II hll2

,

'-I and since both terms of this inequality are continuous in the R-norm, we can eliminate the assumption that h is in A(n)n. 0

Returning to the proof of the theorem, we can apply the lemma to conclude that IIUo - ho)(z)lI~n :5 Ilk .. Jllk ... IIUo - h o)1I2 :5 1/e2 + l/N. Therefore, we see that the vector IN = 10 - hN in Rn satisfies

(1) Tt>UN - hN) = I,


(2) 111 N - hN 11~ ::; 1/(;2 + l/N and (3) 11(1 N - hN )(~)11~ .. ::; 1/(;2 + 1/ N for i ::: 1"", N.

Since the sequence {! N }~=1 in 'Rn is uniformly bounded in norm, there exists a subsequence converging in the weak°-topology to a vector 1/J in 'R"'. Since weak*-convergence implies pointwise convergence, we see that E; 1 IfIj1/JJ = 1 and

"",(Zt) ~ .. ::; 1 e2 for all ~. Since 1/J is continuous on n and the set {z,} is dense in n, it follows that 1/1 is in He;. (n) and 111/J11 ::; 1/(;2 which concludes the proof. 0

Note that we conclude that 1/1 is in wx>(n) and not in M which would be the hoped for

One can note that the argument involving the min-max theorem enables one to show that there are vectors h in K. which satisfy

1 ( 2 1 1 1 kz. f - h) 11 ::; (;2 + N'

Moreover, this shows that there are vectors; so that T~i = 1, 11;11 2 ::; 1/e;2 + l/N, and i~) 2 ::; 1 (;2 + l/N for i = 1, ... ,N. An easy compactness argument completes the proof since the sets of vectors for each N are convex, compact and nest.ed ADd hence have a point in common. 0

5. Concluding comments

With the definitions given, the question arises of which Hilbert modules are (MAP r which quasI-free ones are WMAP. Lemma 4 combined with observations in 23] show that both H2(Bm) and H2(][J)m) are WMAP. Indeed any L~ space f r a measure supported on aBm or the distinguished boundary of ][J)m has these properties. One could also ask for which quasi-free Hilbert modules 'R the kernel functions {k .. }zeo are in M and whether the Toeplitz operators Tk. are invertible

perators as they are in the cases of H2(Bm) and H2(][J)m). It seems possible that the kernel functions for all quasi-free Hilbert modules might have these properties when n is strongly pseudo-convex, with smooth boundary. In many concrete cases, the kzo are actually holomorphic on a neighborhood of the closure of n for Zo in n, where the neighborhood, of course, depends on zoo

Note that the formulation of the criteria in terms of a cyclic submodule S of the quasi-free Hilbert modules makes it obvious that the condition

TS ITS). > e;2 I ~\ ~ _ S

is equivalent to T~T: ~ e2 In.

n the generating vector for S is a cyclic vector. This is Theorem 2 of t231. Also it is easy to that the assnmption on the Toeplitz operators for all cyclic 8ubmodules is equivalent to assuming it for all submodules. That is because

I(Ps ® Ic" )T:/II ~ lI(p[/J ® Icn )T:fll

for f in the submodule S. H for the ball or polydisk we knew that the function "representing" the modulus

of a vector-valued function could be taken to be continuous on clos(n) or cyclic, the corona problem would be solved for those No such result is known, however, and it seems likely that such a result is false.

88 R.G.DOUGLASANDJ.SARKAR

Finally, one would also like to reach the conclusion that the function ,p is in the multiplier algebra even if it is smaller than HOO(O). In the recent paper [91 of Costea, Sawyer and Wick this goal is achieved for a family of spaces which includes the Drury-Arveson space. It seems possible that one might be able to modify the line of proof discussed here to involve derivatives of the {<P'}~l to accomplish this goal in this case, but that would clearly be more difficult.

References

1. M. B. Abrahamse and R. G. Douglas, A class of su/mormal operatqrs related to mulbplyconnected domams, Advances in Math. 19 (1976), no. 1, 106 148.

2. J. Agler and J. E. McCarthy, Nevanlinna Ptck mterpolatton on the btmsk, J. Reine Angew Math. 606 (1999), 191-204.

3. E. Amar, On the Toeplitz corona problem, Pub!. Mat. 47 (2003), no. 2, 489-496. 4. C.-G. Ambrozie and D. Timotin, On an intertwimng ltftmg theorem fur certmn reproducmg

kernel Hilbert spaces, Integral Equations Operator Theory 42 (2002 , no. 4, 313 384-5. M. Andersson and H. Carlsson, Estimates of solutt01lB of the HP and BMOA wrona problem,

Math. Ann. 316 (2000), no. 1,83-102. 6. W. Arveson, Interpolation problems in nest algebras, J. Functional Analysis 20 1975, no. 3,

208-233. 7. J. A. Ball, T. T. Trent, and V. Vinnikov, Interpolation and commutant bftmg far multlplters

on reproducing kernel Hilbert spaces, Operator Theory and Analysis Amsterdam. 1997 , Oper. Theory Adv. Appl., vol. 122, Birkhauser, Basel, 2001, pp. 89-138.

8. L. Carleson, Interpolations by bounded analytic juncti01lB and the corona problem, Ann. f Math. (2) 76 (1962),547-559.

9. S. Costea, E. T. Sawyer, and B. D. Wick, The corona theorem for the Drury-Arneson Bani space and other holomorphic Besov - Sobolev spaces on the umt ball m en, available at arIl.'9: 0811.0627.

10. K. R. Davidson and R. G. Douglas, The genemlized Berenn tronsform and commutatortdeaLs, Pacific J. Math. 222 (2005), no. 1,29-56.

11. R. G. Douglas, On majorization, factorization, and range mclUMon of operators on Bllberi space, Proc. Amer. Math. Soc. 17 (1966),413-415.

12. ___ , Banach algebm techniques in opemtor theory, Vol. 49, Academic Press, New York, 1972. Pure Appl. Math.

13. R. G. Douglas and G. Misra, On quasi-free Hilbert modules, New York J. Math. 11 2005, 547-561.

14. R. G. Douglas and V. 1. Paulsen, Hilbert modules over junchon algebras, Pitman Res. Notes Math. Ser., vo!. 217, Longman Sci. Tech., Harlow, 1989.

15. J. Eschrneier and M. Putinar, Sphencal contmcti01lB and \nterpoiahon problems on the unst ball, J. Reine Angew. Math. 542 (2002),219 236.

16. T. W. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, NJ, 1969. 17. S.-Y. Li, Corona problem of several comple:l: vanables, The Madison Symposium on Complex

Analysis (Madison, WI, 1991), Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 307 - 328.

18. K.-C. Lin, HP-801uh01lB for the corona problem on the polyd\Sc \n en, Bull. Sci. Math. (2) 110 (1986), no. 1, 69 84.

19. B. Sz.-Nagy and C. Foias, Harmomc analy81S of opemtors on Ht/bert space, North-Holland, Amsterdam, 1970.

20. C. F. Schubert, The corona theorem as an opemtor theorem, Proc. Amer. Math. Soc. 69 (1978), no. 1, 73 76.

21. S. Treil and B. D. Wick, The matrix-valued HP corona problem tn the d\Sk and polyd\8k, J. Funct. Anal. 226 (2005), no. 1, 138 172.

22. T. T. Trent, Soluti01lB for the H""(Dn) corona problem belonging to exp(Ll/(2n-l»), Recent Advances in Matrix and Operator Theory, Oper. Theory Adv. Appl., vol. 179, Birkhauser, Basel, 2008, pp. 309 328.

23. T. T. Trent and B. D. Wick, Toeplit.z corona theorems for the polyd\Sk and the umt bal~ Complex Anal. Oper. Theory 3 (2009), no. 3, 729 738.

SOME REMARKS ON THE TOEPLITZ CORONA PROBLEM

24. N. Th. Varopoul08, Probabtlisttc approach to Bome problems in cornpla anai1lBtB, Bull. Sci. Math. (2) 105 (1981), no. 2, 181 224.

DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STATION, TX 77843-

3368, USA E-mail address, R. G. Douglas: rdouglasGmath.tamu.edu E-mail address, J. Sarka!': j sarkarGmath. tamu. edu

Contre de Reehereh811 M .. th'matlqu811 CRM Proceedings and Lecture Not .. Volume Ill. ~OlO

Regularity on the Boundary in Spaces of Holomorphic Functions on the Unit Disk

Emmanuel Fricain and Andreas Hartmann

ABSTRACl'. We review some results on regularity on the boundary in spaces of analytic functions on the unit disk connected with backward shift invariant subspaces in EfP.

1. Introduction

Fa.tou's theorem shows that every function of the Nevanlinna class N := {J E HoI D : sUPO<r<l f':w 10g+lf(reit)1 dt < oo} admits nontangentiallimits at almost every point ( of the unit circle T = aD, D = {z E IC : Izl < I} being the unit disk. One can easily construct functions (even contained in smaller classes) which do not admit nontangentiallimits on a dense set of T. The question that arises from such an observation is whether one ca.n gajn regularity of the functions at the boundary when restricting the problem to interesting subclasses of N. We will discuss two kinds of subclasses corresponding to two different ways of generalizing the class of st.andard backward shift invariant subspaces in H2 := {f E Hol(D) : f ~ := limr-+l 1/27r) J':wlf(reitW dt < oo}. Recall that backward shift invariant

subspaces have shown to be of great interest in many domains in complex analysis and operator theory. In H2, they are given by K'f := H2 e I H2 , where I is an inner function, that is a bounded analytic function in D the boundary values of which are in modulus equal to 1 a.e. on T. Another way of writing K'f is

K'f = H2 n IH~, where m = z}{l is the subspace of functions in H2 vanishing in O. The bar sign means complex conjugation here. This second writing K'f = H2 n I H~ does not appeal to the Hilbert space structure and thus generalizes to HP (which is defined as IJ2 but replacing the integration power 2 by p E (0,00); it should be noted that for p E (0,1) the expression IIfll~ defines a metric; for p = 00, HOC is the Banach sp~ of bounded analytic functions on D with obvious norm). When p = 2, then

2000 MathematU:8 SubJect CI.aB81jication. 30B40, 30D66, 47 A16, 47B37. Ketl words and phrasell. analytic continuation, backward shift, de Branges- Rovnyak spaces,

Carleson E.IIIbedding, connected level set condition, invariant subspace, Muckenhoupt condition, reproducing kemels, rigid functions, spectrum, Toeplitz operators.

This is the fira' fOim of the paper.

@2010 Am.rlc .... M .. thom .. tlc .. 1 Society

91

92 E. FRICAIN AND A. HARTMANN

these spaces are also called model spaces because they arise in the construction of a universal model for Hilbert space contractions developped by Sz.-Nagy Foias (see [71]). Note that if 1 is a Blaschke product associated with a sequence (Zn n>l of points in ID>, then Kf coincides with the closed linear span of simple fractions-with poles of corresponding multiplicities at the points l/zn.

Many questions concerning regularity on the boundary for functions in standard backward shift invariant subspaces were investigated in the extensive existing literature. In particular, it is natural to ask whether one can find points in the boundary where every function f in Kf and its derivatives up to a given order have nontangential limits; or even can one find some arc on the boundary where every function f in Kf can be continued analytically? Those questions were investigated by Ahern - Clark, Cohn, Moeller, .... Another interest in backward shift invariant subspaces concerns embedding questions, especially when Kf embeds into some V(J.L). This question is related to the famous Carleson embedding theorem and was investigated for instance by Aleksandrov, Cohn, Treil, Volberg and many others (see below for some results).

In this survey, we will first review the important results in connection with regularity questions in standard backward shift invariant subspaces. Then we ",ill discuss these matters in the two generalizations we are interested in: de BrangesRovnyak spaces on the one hand, and weighted backward shift invariant subspaces-which occur naturally in the context of kernels of Toeplitz operatorson the other hand. Results surveyed here are mainly not followed by proofs. However, some of the material presented in Section 4 is new. In particular Theorem 18 for which we provide a proof and Example 4.1 that we will discuss in more detail. The reader will notice that for the de Branges - Rovnyak situation there now exists a quite complete picture analogous to that in the standard Kf spaces whereas the weighted situation has not been investigated very much yet. The example 4.1 should convince the reader that the weighted situation is more intricate in that the Ahern-Clark condition even under strong conditions on the weight-that ensure, e.g., analytic continuation off the spectrum of the inner function - is not sufficient.

2. Backward shift invariant subspaces

We will need some notation. Recall that the spectrum of an inner function 1 is defined as 0"(1) = {( E closID>: liminf",-+( l(z) = O}. This set corresponds to the zeros in ID> and their accumulation points on T = 8ID>, as well as the closed support of the singular measure J.Ls of the singular factor of [.

The first important result goes back to Moeller [56] (see also [1] for a several variable version):

Theorem 1 (Moeller, 1962). Let r be an open arc ofT. Then every function f E Kf can be continued analytically through r if and only if r n O"(I) = 0.

Moeller also establishes a link with the spectrum of the compression of the backward shift operator to Kf.

It is of course easy to construct inner functions the spectrum of which on l' is equal to T 80 that there is no analytic continuation possible. Take for instance for 1 the Blaschke product associated with the sequence A = {(1-1/n2)ein}n, the zeros of which accumulate at every point on T. So it is natural to ask what happens in points which are in the spectrum, and what kind of regularity can be expected there.

REGULARITY ON THE BOUNDARY 93

Ahern Clark and Cohn gave an answer to this question in [2,28]. Recall that an arbitrary inner function I can be factored into a Blaschke product and a singular inner function: 1= BS, where B = TIn btln , btln (z) = (lanl/anHan - z)/(1- anz), L:n(1-lanI2) < 00, and

S(z) = exp -(+z ,. d/LS «) ,

T.,,-z

where /Ls is a finite positive measure on 'll' singular with respect to normalized Lebesgue measure m on 'll'. The regularity of functions in Kf is then related with the zero distribution of B and the measure /LS as indicated in the following result.

Theorem 2 (Ahern - Clark, 1970, Cohn, 1986). LetI be an inner function and let 1 < p < +00 and q its conjugated exponent. If I is a nonnegative integer and ( E'll', then the following are equivalent:

1

(i) for every f in Kf, the functions f(j), 0 $ j $ I, have finite nontangential ltmtts at (;

(ii we have S;(l+l)«) < +00, where

1 it r ... =1 11- (anl T 0 11- (eitlr d/Ls(e )

(1$r<00).

Moreover m that case, the function (kDI+1 belongs to KJ and we have

---;c-:-:-Il+l 2 f(l)«() = I! ilf(z)k~(z) dm(z),

T

for every function f E Kf. Here k~ is the reproducing kernel of the space K; corresponding to the point

( and defined by

3 k~(z) = 1- I«(~I(z). 1-(z

The quantity S; «) is closely related to the angular derivatives of the inner function 1. Recall that a holomorphic selfmap f of the unit disk JI)) is said to have an angular derivative at ( E 'll' if f has nontangential limit of modulus 1 in ( and f ( := )jmr~l f'(r() exists and is finite. Now, in the case where f = I is an inner function, if SH () < +00, then I has an angular derivative at ( and SJ( () = II' «) I

[4, Theorem 2]). Moreover, if S{+1 «) < +00, then I and all its derivatives up to order I have finite radial limits at ( (see [3, Lemma 4]).

Note that the case p = 2 of Theorem 2 is due to Ahern - Clark and Cohn . the result to p > 1 (when 1 = 0). Another way to read into the results

of Ahern-Clark, Cohn and Moeller is to introduce the representing measure of the inner function I, /LI = /Ls + /LB, where

/LB := ~(1 -lan I2)8{tln}. n>1 -

Then Theorems 1 and 2 allow US to formulate the following general principle: if the measure /LI is "small" near a point, E 'll', then the functioIlB J in Kf must be smooth near that point.

Another type of regularity questioIlB in backward shift invariant subspaces was studied by A. Aleksandrov, K. Dyakonov and D. KhaviIlBon. It cOIlBists in asking if

94 E. FRlCAIN AND A. HARTMANN

Kf contains a nontrivial smooth function. More precisely, Aleksandrov in [51 proved that the set of functions f E Kf continuous in the closed unit disc is dense in J<1f. It should be noted nevertheless that the result of Aleksandrov is not constructive and indeed we do not know how to construct explicit examples of functions fEW; continuous in the closed unit disc. In the same direction, Dyakonov and Khavinson, generalizing a result by Shapiro on the existence of Cl-functions in K'j [68], proved in [421 that the space KJ contains a nontrivial function of class Aoo if and only if either I has a zero in ][)) or there is a Carleson set E C T with ps(E > 0; here Aoo denotes the space of analytic functions on ][)) that extend continuously to the closed unit disc and that are Coo (1[') ; recall that a set E included in T is said to be a Carleson set if the following condition holds

~ log dist( (, E) dm( () > -00.

In [34,36,37,40]' Dyakonov studied some norm inequalities in backward shift invariant subs paces of HP (C+ ) ; here HP (C+) is the Hardy space of the upper half-plane C+ := {z E C : 1m z > O} and if 8 is an inner function for the upper half-plane, then the corresponding backward shift invariant subspace of HP C+ is also denoted by K~ and defined to be

K~ = HP(C+) n 8HP(C+).

In the special case where 8(z) = eia.z (a > 0), the space Kt is equal to PwP n HP(C+), where pwg is the Paley- Wiener space of entire functi ns of exponential type at most a that belong to £P on the real axis. Dyakonov shows that several classical regularity inequalities pertaining to pwg apply also to Kt provided 8' is in HOO(C+) (and only in that case). In particular, he proved the following result.

Theorem 3 (Dyakonov, 2000 and 2002). Let 1 < p < +00 and let 8 be an inner function in HOO(C+). The following are equivalent:

(4)

(i) K~ C Co(lR). (ii) K~ C Lq(lR), for some (or all) q E (p,+oo). (iii) The differentiation operator is bounded as an operator from ~ to VCR ,

that is

IIf'lIp ~ C(p, 8)lIfllp, (iv) 8' E HOO(C+).

JEK~.

Notice that in (4) one can take C (p, 8) = C1 (P) 8' 00, where C 1 (P) depends only on p but not on 8. Moreover, Dyakonov also showed that the embeddings in (i), (ii) and the differentiation operator on K~ are compact if and only if 8 satisfies (iv) and 8'(x) -+ 0 as Ixl -+ +00 on the real line. In [38], the author discusses when the differentiation operator is in Schatten von Neumann ideals. Finally in [401, Dyakonov studied coupled with (4) the reverse inequality. More precisely, he characterized those 8 for which the differentiation operator f t-+ f' provides an isomorphism between K~ and a closed subspace of HP, with 1 < p < +00; namely he showed that such 8's are precisely the Blaschke products whose zero-set lies in some horizontal strip {a < 1m z < b}, with 0 < a < b < +00 and splits into finitely many separated sequences.

The inequality (4) corresponds for the case 8(z) = eia.z to a well-known inequality of S. Bernstein (see [17, Premier lemme, p. 75] for the case p = +00 and


[19, Theorem 11.3.3] for the general case). For p = +00, a beautiful generalization of Bernstein's inequality was obtained by Levin [55]: let x E IR and le'(x)1 < +00; then for each I E Ke I the derivative f' (x) exists in the sense of nontangential boundary values a.nd

f'(x) < II I e'(x) - 1100, f EKe·

Recently, differentiation in the backward shift invariant subspaces K~ was studied extensively by A. Bara.nov. In [11,13], for a general inner function 6 in HOO(C+), he proved estimates of the form

(5) II/(1)wp ,I\\LP(I') ::; Glllllp, f E K~,

where I ~ 1, p. is a Carleson measure in the closed upper half-plane and Wp,1 is some weight related to the norxn of reproducing kernels of the space K~ which compensates for possible growth of the derivative near the boundary. More precisely, put

Wp,I(Z) = n(k~)l+ll1;p/(p+l), (z E clos(C+)),

where q is the conjugate exponent of p E [1,+00). We assume that Wp,!(x) = 0, whenever S~I+l (x) = +00, x E IR (here we omit the exact formula of k~ and S~ in the upper half-plane but it is not difficult to imagine what will be the analogue of 1 a.nd 3) in that case).

Theorem 4 (Baranov, 2005). Let p. be a Carleson measure in clos(C+), lEN, 1 ::; p < +00. Then the operator

(Tp,d)(z) = f(!)(z)wp,!(z)

IS f weak type (p, p) as an operator from K~ to V (p.) and is bounded as an operator from K9 to Lr p.) lor any r > p; moreover there is a constant C = C(p., p, r, l) such that

f EKe·

The proof of Baranov's result is based on the integral representation (2) which the study of differentiation operators to the study of certain integral singular

perators. To apply Theorem 4, one should have effective estimates of the considered

weights, that is, of the norms of reproducing kernels. Let

fl(6,e):= {z E C+: 18(z)1 < e}

be the level sets of the inner function 6 and let de(x) = dist(x,!1(8,e)), x E IR. Then Baranov showed in [12] the following estimates:

6 ~(x) :s Wp,!(x) :s 18'(x)I-I, x E IR.

Using a result of A. Aleksandrov [8], he also proved that for the special class of jnner functions e satisfying the connected level set condition (see below for the definition in the framework of the unit disc) and such that 00 E u(6), we have

7) wpl(x)xle'(x)r l (xEIR).

In fact, the inequalities (6) and (7) are proved in [12, Corallary 1.5 and Lemma 4.5] for I = 1; but the argument extends to generall in an obvious way. We should mention that Theorem 4 implies Theorem 3 on boundeness of differentiation operator. Indeed if 8' E LOD(IR), then it is clear (and well known) that sUP3:elltllk~lIq < +00,

96 E. FRICAIN AND A. HAIrrMANN

for any q E (1,00). Thus the weights Wr = Wr,l are bounded from below and thus inequality

(fe~) implies inequality (4).

Another type of results concerning regularity on the boundary for functions in standard backward shift invariant subspaces is related to Carleson's embedding theorem. Recall that Carleson proved (see [21,221) that H'P = HP(]!} embeds continuously in V(Ji.) (where Ji. is a positive Borel measure on clos ]!})) if and only if Ji. is a Carleson measure, that is there is a constant G = G(I') > 0 such that

Ji.(S«(, h)) $ Gh,

for every "square" S«(, h) = {z E clos]!} : 1 - h/21r $ \z\ $ 1, arg(z( $ h 2}, ( E '][', hE (0,21r). The motivation of Carleson comes from interpolation problems but his result acquired wide importance in a larger context of singular integrals of Calderon - Zygmund type. In [271, Cohn studied a similar question for model subspaces KJ. More precisely, he asked the following question: given an inner function I in JI)) and p ~ 1, can we describe the class of positive Borel measure p in the closed unit disc such that Kf is embedded into l.J'(I')? In spite of a number of beautiful and deep (partial) results, this problem is still open. Of course, due to the closed graph theorem, the embedding Kf c l.J'(Ji.) is equivalent to the estimate

(8) II/IILP(J') $ Gll/ilp (f e Kf)· Cohn solved this question for a special class of inner functions. \Ve recall that I is said to satisfy the connected level set condition (and we write leG LS if the level set n(I, c) is connected for some c e (0,1).

Theorem 5 (Cohn, 1982). Let Ji. be a positive Borel measure on closJl). Let I be a an inner function such that lEG LS. The following are equ valent:

(i) KJ embedds continuously in L2(Ji.). (ii) There is c > ° such that

(9) 1 1-\z\2 G cloBJI) \1 - z(12 dJi.«) $ 1 -ll(z)12'

z eJI)).

It is easy to see that if we have inequality (8) for f = k~ z e 11)), then we have inequality (9). Thus Cohn's theorem can be reformulated in the following way: inequality (8) holds for every function f e KJ if and only if it holds for reproducing kernels f = k~, z E [J). Recently, F. Nazarov and A. Volberg [58] showed that this is no longer true in the general case. We should compare this property of the embedding operator KJ C L2(1') (for CLS inner functions) to the "reproducing kernel thesis," which is shared by Toeplitz or Hankel operators in H2 for instance. The reproducing kernel thesis says roughly that in order to show the boundeness of an operator on a reproducing kernel Hilbert space, it is sufficient to test its boundeness only on reproducing kernels (see, e.g .. [59, YoU, pp. 131,204, 244, 246] for some discussions of this remarkable property).

A geometric condition on JJ sufficient for the embedding of Kf is due to Volberg 'Treil [73].

Theorem 6 (Volberg Treil, 1986). Let I' be a positive Borel measure 011

closJl), let I be a an inner function and let 1 $ P < +00. Assume that there IS


c > 0 such that

(10) JL(S«, h)) < Ch,

for every square S«, h) satisfying S«, h) n O(I,e) :f. 0. Then Kr embeds continuously in V'(JL).

Moreover they showed that for the case where I satisfies the connected level set condition, the sufficient condition (10) is also necessary, and they extend Theorem 5 to the Banach setting. In [8], Aleksandrov proved that the condition of Volberg Treil is if and only if IE CLS. Moreover, if I does not satisfy the connected level set condition, then the class of measures JL such that the inequality (8) is valid depend essentially on the exponent p (in contrast to the classical theorem of Carleson).

Of special interest is the case when JL = }:nEN anb'p.n} is a discrete measure; then embedding is equivalent to the Bessel property for the system of reproducing b.-ernels {ki }. In fact, Carleson's initial motivation to consider embedding properties comes from interpolation problems. These are closely related with the Ries2 basis property which itself is linked with the Bessel property. The Riesz basis propertyofreproducing kernels {kIn} has been studied by S. V. HruScev, N. K. Nikol'skil and B. S. Pavlov in the famous paper [51], see also the recent papers by A. Baranov 13,14] and by the first author [23,43]. It is of great importance in applications

such as f r instance control theory (see [59, Vol. 2]). Also the particular case when JL is a measure on the unit circle is of great inter

est. In contrast to the embeddings of the whole Hardy space HP (note that Carleson measures n 'll' are measures with bounded density with respect to Lebesgue mea.sure , the class of Borel measures JL such that Kf C V'(JL) always contains n ntnvial examples of singular measures on 'll'j in particular, for p = 2, the Clark

26] for which the embeddings KJ C L2 (JL) are isometric. Recall that gl\ en >. E 'll', the Clark measure 0). associated with a function b in the ball of HOO

is defined as the unique positive Borel measure on 'll' whose Poisson integral is the real part of >. + b)/(>' - b). When b is inner, the Clark measures OQ are singular with to the Lebesgue measure on 'll'. The situation concerning embeddings f r Clark measures changes for p :f. 2 as shown by Aleksandrov [6J: while for p 2:: 2 this embedding still holds (see [6, Corollary 2, p. 117]), he constructed an example f r hich the embedding fails when p < 2 (see [6, Example, p. 123]). See also the nice survey by Poltoratski and Sarason on Clark measures [60J (which they call Aleksandrov-Clark measures). On the other hand, if JL = wm, w E L2(']['), then the embedding problem is related to the properties of the Toeplitz operator Tw (see 29) .

In [11,12], Baranov developped a new approach based on the (weighted norm) in inequalities and he got BOrne extensions of Cohn and Volberg Treil re

sults. Compactness of the embedding operator Kf C V'(JL) is also of interest and is considered in [12,15,24,29,72].

Another important result in connection with Kf-spaces is that of Douglas, Shapiro and Shields ([32), see also [25, Theorem 1.0.5; 62)) and concerns pseudo-

• continuation. Recall that a function holomorphic in]]])e := C \ clos]]]) clos E means the closure of a set E is a pseudocontinuation of a function f meromorphic in ]() if 'I/J vanishes at 00 and the outer nontangential limits of 'I/J on '][' coincide with the inner nontangential limits of J on l' in almost every point of ']['. Note that


I E KJ = H2 n I Hg implies that I = 17jj with "" E ~. Then the meromorphic function I I I equals 1jj a.e. T, and writing 1/J(z) = ~">1 b .. z", it is clear that

.if;(z) := En~l bnlzn is a holomorphic function in De, vani;hing at 00, and being

equal to III almost everywhere on T (in fact,.if; E .H2(ll)e». The converse is also true: if III has a pseudocontinuation in ll)e, where f is a HI'-function and I some inner function I, then I is in Kf. This can be resumed this in the following result.

Theorem 7 (Douglas-Shapiro-Shields, 1972). Let I be an tnner /unction. Then a function I E HI' is in Kf il and only il I I I has a pseudocont nuahon to a function in HI'(ll)e) which vanishes at infinity.

Note that there are functions analytic on C that do not admit a pseudocontinuation. An example of such a function is I(z) = e" which has an essential singularity at infinity.

As already mentioned, we will be concerned with two generalizati ns of the backward shift invariant subspaces. One direction is to consider weighted versions of such spaces. The other direction is to replace the inner function by m re general functions. The appropriate definition of KJ in this setting is that of de BrangesRovnyak spaces (requiring that p = 2).

Our aim is to discuss some of the above results in the context of these spaces. For analytic continuation it turns out that the conditions in both cases are quite similar to the original KJ-situation. However in the weighted situation some additional condition is needed. For boundary behaviour in points in the spectrum the situation changes. In the de Branges - Rovnyak spaces the Ahern - Clark condition generalizes naturally, whereas in weighted backward shift invariant subspaces the situation is not clear and awaits further investigation. This will be illustrated in Example 4.1.

3. de Branges - Rovnyak spaces

Let us begin with defining de Branges-Rovnyak spaces. We will be essentially concerned with the special case of Toeplitz operators. Recall that for f{) E Loo l' , the Toeplitz operator T cp is defined on H2 by

where P+ denotes the orthogonal projection of L2('][') onto H2. Then, for VJ E Loo('Jl'), I\VJl\oo ::; 1, the de Branges Rovnyak space 1-£(f{), associated with f{), consists of those H2 functions which are in the range of the operator (Id - TcpT~)l 2.

It is a Hilbert space when equipped with the inner product

(Id - TcpTcp) 1/2 I, (Id - TcpTcp)l 2g)", = (f,gh,

where I, 9 E H2 e ker(Id - TcpTcp) 1/2 •

These spaces (and more precisely their general vector-valued version) appeared first in L. de Branges and J. Rovnyak [30,31] as universal model spaces for Hilbert space contractions. As a special case, when b = I is an inner function (that is Ibl == III = 1 a.e. on 'Jl'), the operator (Id - TIT1 ) is an orthogonal projection and 11.(1) becomes a closed (ordinary) subspace of H2 which coincides with the model spaces K1 = H2 e I H2. Thanks to the pioneering work of Sarason, e.g., [64-67], we know that de Branges Rovnyak spaces play an important role in numerous questions of complex analysis and operator theory. We mention a recent paper by the second

REGULARI'l'Y ON THE BOUNDARY 99

n8.TDed author and Sarason and Seip [47] who gave a characterization of surjectivity of Toeplitz operator the proof of which involves de Branges-Rovnyak spaces. We also refer to work of J. Shapiro [69,70] concerning the notion of angular derivative for holomorphic self-maps of the unit disk. See also a paper of J. Anderson and J. Rovnyak [10], where generalized Schwarz Pick estimates are given and a paper ofM. Jury [52], where composition operators are studied by methods based on 1£(b) spaces.

In what follows we will assume that b is in the unit ball of Hoo. We recall here that since 1£(b) is contained contractively in H2, it is a reproducing kernel Hilbert space. l\lore precisely, for all function f in 1£(b) and every point>. in lDl, we have

11) f(>.) = (1, k~h,

where k~ = (Id - T"TII)kA• Thus

k~ (z) = 1 - b(A2b(z) , 1- >.Z

z E lDl.

We also lecall that 1£(b) is invariant under the backward shift operator and in the following, we denote by X the contraction X := S;1£(b)' Its adjoint satisfies the import-ant formula

X*h = Sh - (h, S*bhb, hE 1£(b).

In the case where b is inner, then X coincides with the so-called model operator of Sz.-Nagy-Foias which serves as a model for certain Hilbert space contractions (in fact, those contractions T which are 0.0 and with Or = Or. = 1; for the general case, the model operator JS quite complicated).

Finally, let us recall that a point>. E ii} is said to be regular (for b) if either A e ]) and b A) =I- 0, or .>. E 'f and b admits an analytic continuation across a neIghbourhood VA = {z : Iz -'>'1 < €} of .>. with Ibl = 1 on VA n 'f. The spectrum of b, den ted by u(b), is then defined as the complement in ii} of all regular points of b. F r the case where b = I is an inner function, this definition coincides with the definition given before.

In this section we will summarize the results corresponding to Theorems 1 and 2 above in the setting of de Branges-Rovnyak spaces. It turns out that Moeller's result remains valid in the setting of de Branges- Rovnyak spaces. Concerning the lCsult by Ahem-Clark, it turns out that if we replace the inner function I by a general function b in the ball of Hco, meaning that b = Ibo where bo is now outer, then we have to add to condition (ti) in Theorem 2 the term corresponding to the absolutely continuous part of the measure: Iloglboll.

In (441, the first named author and J. Masbreghi studied the continuity and analyticity of functions in the de Branges - Rovnyak spaces 1£ (b) on an open arc of T. AB we will see the theory bifurcates into two opposite cases depending on whether b is an extreme point of the unit ball of Hoo or not. Let us recall that if X is a linear space and S is a convex subset of X, then an element xeS is called an extreme point of S if it is not a proper convex combination of any two distinct points in S. Then, it is well known (see [33, p. 125]) that a function f is an extreme point of the unit ball of Hoo if and only if

log(1 -If(()I) dm«) = -00. T


The following result is a generalization of Theorem 1 of Moeller.

Theorem 8 (Sarason 1995, Fricain-Mashreghi, 2008). Let b be in the unit ball of HOO and let r be an open arc of 1'. Then the following are equivalent:

(i) b has an analytic continuation across rand b = 1 on r; (ii) r is contained in the resolvent set of x·; (iii) any function f in 1l(b) has an analytic continuation across r; (iv) any function f in 1l(b) has a continuous extension to Jl}) U rj (v) b has a continuous extension to Jl}) U rand Ibl = 1 on r.

The equivalence of (i), (ii) and (iii) were proved in [67, p. 421 under the assumption that b is an extreme point. The contribution of Fricain - Mashreghi concerns the last two points. The mere assumption of continuity implies analyticity and this observation has interesting application as we will see below. Note that this implication is true also in the weighted situation (see Theorem 18 .

The proof of Theorem 8 is based on reproducing kernel of 1l b) spaces. More precisely, we use the fact that given wE lIJ), then k~ = (Id - wX· -lk" and thus

for every f E 1l(b). Another key point in the proof of Theorem 8 is the theory of Hilbert spaces contractions developped by Sz.-Nagy-Foias. Indeed, if b is an extreme point of the unit ball of Hoo, then the characteristic function of the contraction X· is b (see (631) and then we know that O'(X*) = 0' b .

It is easy to see that condition (i) in the previous result implies that b is an extreme point of the unit ball of Hoo. Thus, the continuity or equivalently, the analytic continuation) of b or of the elements of 1l(b) on the boundary completely depends on whether b is an extreme point or not. If b is not an extreme point of the unit ball of Hoo and if r is an open arc of 1', then there exists necessarily a function f E 1l(b) such that f has not a continuous extension to lIJ) u r. On the opposite case, if b is an extreme point such that b has continuous extension to lIJ)Ur with \b\ = 1 on r, then all the functions f E 1l(b) are continuous on r (and even can be continued analytically across r).

As in the inner case (see Ahern-Clark's result, Theorem 2), it is natural to ask what happens in points which are in the spectrum and what kind of regularity can be expected there. In [441, we gave an answer to this question and this result generalizes the Ahem - Clark result.

Theorem 9 (Fricain Mashreghi, 2008). Let b be a point in the unit ball of HOO and let

(12) b(z) = 'Y n(lanl an - z ) exp (-l ; + z d~()) exp (l ; + z log b() dm()) n an 1 - anz T .. - Z T .. - Z

be its canonical factorization. Let (0 E l' and let l be a nonnegative integer. Then the following are equivalent.

(i) each function in 1-£(b) and all its derivatives up to order l have (fimte) radial limits at (0;

(ii) IIQlk~/azlllb is bounded as z tends radially to (0; (iii) X· I k~ belongs to the range of (Id - (oX·)1+1;


(iv) we have S~I+2«O) < +00, where

2,.. Iloglb(e1t)l\ d (It)

1 'I me, o (0 - elt r

1 -lan l2 2,.. d/L(elt )

(1 ~ r < +00),

In the following, we denote by Er(b) the set of points (0 E 'll' which satisfy S~«o) < +00.

The proof of Theorem 9 is based on a generalization of technics of Ahern Clark. However, we should mention that the general case is a little bit more complicated than the inner case. Indeed if b - 1 is an inner function, for the equivalence of iii) and (iv) (which is the hard part of the proof), Ahern Clark noticed that the

condition (ill) is equivalent to the following interpolation problem: there exists k, 9 E H2 such that

(1- (oz)l+1k(z) -l!zl = I(z)g(z).

This reformulation, based on the orthogonal decomposition H2 = 11.(1) EEl IH2, is crucial in the proof of Ahern Clark. In the general case, this is no longer true because li(b) is not a closed subspace of H2 and we cannot have such an orthogonal decomposition. This induces a real difficulty that we can overcome using other arguments: in particular, we use (in the proof) the fact that if (0 E EI+1(b) then, for 0 ~ j ~ l, the limits

lim bej) (r(o) and lim b(j) (R(o) r--+l- R--+l +

exist and are equal (see 13]). Here by reflection we extend the function b outside the unit disk by the formula (12), which represents an analytic function for Izl > 1, z i= 1 an. We denote this function also by b and it is easily verified that it satisfies

1 13 b(z) - V z E C.

- b(l/z) '

Maybe we should compare condition (iii) of Theorem 9 and condition (ii) of Theorem 8. For the question of analytic continuation through a neighbourhood V<o of a point (0 E 'll', we impose that for every z E V<o nT, the operator Id - zX· is bijective (or onto which is equivalent because it is always one-to-one as noted in [43, Lemma 2.2]) whereas for the question of the existence of radial limits at (0 for the derivative up to a given order l, we impose that the range of the operator Id - (oX*)l+l conta.ins the only function X*I k~. We also mention that Sarason

has obtained another criterion in terms of the Clark measure (1). associated with b above for a definition of Clark measures; note that the Clark measures here

are not always singular as they are when b is inner).

Theorem 10 (Sarason, 1995). Let (0 be a point ofT and let l be a nonnegatwe integer. The followmg conditions are equivalent.

(i) Each function in li(b) and all its derivatives up to order I have nontangential limits at (0.

(ii) There is a point oX E T such that

(14) lei6 - (01-21 - 2 du>.(eI6

) < +00. T

(iii) The last inequality holds for all oX E 'll' \ {b«o)}·


(iv) There is a point>. E T such that IJ). has a point mass at (0 and

{ \ei9 _ (0\-21 du).(ei9 ) < 00. JT\{zo}

Recently, Bolotnikov and Kheifets [201 gave a third criterion (in some sense more algebraic) in terms of the Schwarz- Pick matrix. Recall that if b is a function in the unit ball of Hoo, then the matrix P,(z), which will be refered to as to a Schwarz-Pick matrix and defined by

pb(z := r ~ ~+j .1- \b(z)\211

,

I) l~!J!8z'8z3 1-\z\2 ',3=0

is positive semidefinite for every I 2:: 0 and z E lD>. We extend this notion to boundary points as follows: given a point (0 E T, the boundary Schwarz-Pick matrix is

pr((o) = lim pY(z) z--t(o

<I

(I 2:: 0),

provided this nontangentiallimit exists.

Theorem 11 (Bolotnikov-Kheifets, 2006). Let b be a point in the unit ball of Hoo , let (0 E T and let I be a nonnegative integer. Assume that the boundary Schwarz-Pick matrix PY((o) exists. Then each function in 1/. b) and all its derivatives up to order I have nontangential limits at (0'

Further it is shown in [201 that the boundary Schwarz - Pick matrix pr (0 exists if and only if

(15)

where

lim db,I(Z) < +00, z--t(o

<I

1 821 1 - \b(z)\2 db,I(Z) := (1!)28z18z1 1 - \z12 .

We should mention that it is not clear to show direct connections between conditions (14), (15) and condition (iv) of Theorem 9.

Once we know the points (0 in the unit circle where /(1)«(0) exists (in a nontangential sense) for every function / E 1/.(b), it is natural to ask if we can obtain an integral formula for this derivative similar to (2) for the inner case. However, if one tries to generalize techniques used in the model spaces KJ in order to obtain such a representation for the derivatives of functions in l£(b), some difficulties appear mainly due to the fact that the evaluation functional in l£(b) (contrary to the model space KJ) is not a usual integral operator. To overcome this difficulty and nevertheless provide an integral formula similar to (2) for functions in 1/.(b) , the first named author and Mashreghi used in [451 two general facts about the de BrangesRovnyak spaces that we recall now. The first one concerns the relation between 1i.(b) and 1i.(b). For f E H2, we have [67, p. 101

f E 1i.(b) ~ Tr'/ E 1i.(b).

Moreover, if /1,/2 E 1i.(b), then

(16)


We also mention an integral representation for functions in 1-£(b) [67, p. 16}. Let p«() := 1 - Ib«()12 , ( E 'Jl', and let L2(p) stand for the usual Hilbert space of measurable functions I: 'Jl' -+ C with II/lip < 00, where

II/II~ := 1/(()12 p«() dm«(). T

For each A E ID>, the Cauchy kernel k>. belongs to L2(p). Hence, we define H2(p) to be the (closed) span in L2(p) of the functions k>. (A e ID». If q is a function in L2(p), then qp is in L2('Jl'), being the product of qp1/2 E L2(11') and the bounded function pl 2, Finally, we define the operator Cp: L2(p) -+ H2 by

Cp(q) := P+(qp).

Then Cp is a partial isometry from L2(p) onto 1/.(6) whose initial space equals to H2 p) 8.Dd it is 8.D isometry if and only if b is an extreme point of the unit ball of H OO

,

Now let w E closlD> and let I be a nonnegative integer. In order to get an integral representation for the lth derivative of I at point w for functions in the de Branges Rovnya.k spaces, we need to introduce the following kernels

17 b ,_ ,z' - b(z) 'E~=o(b(p)(w)/p!)zl-P(I- wz)P

kw'(z) ,- l. ( _ )1+1 , (z E ID», l-wz ADd

18 «( E 11').

Of course, for w = (0 E 11', these formulae have a sense only if b has derivatives (in a radial or nontangential sense) up to order lj as we have seen this is the case if ( e E'+l b) (which obvlOusly contajns E2(1+l) (b».

F r l = 0, we see that k! 0 = k! is the reproducing kernel of 1-l(b) and k~ 0 = I' , b w k", is (up to a constant) the Cauchy kernel. Moreover (at least formally) the function k~" (respectively k:,I) is the lth derivative of k!,o (respectively of k:,o) with to w.

Theorem 12 (Fricain-Mashreghi, 2008). Let b be a function in the unit ball 01 Hoo and let 1 be a nonnegative integer. Then for every point (0 E lD>UE21+2(b) and lor every function I E 1-£(b), we have k~o,l E 1-l(b), k~o,' e L2(p) and

19) 1(1)«(0) = I«()k~o 1(0 dm«() + g«()p«()k~o,l«() dm«(), T" ']I'

9 e H2(P) satisfies nf = Cpg.

We should say that Theorem 12 (as well as Theorem 13, Proposition 1, Thee? rem 14, Theorem 15 and Theorem 16 below) are stated and proved in [16,45] in the framework of the upper half-plane; however it is not difficult to see that the same recbnies can be adapted to the unit disc and we give the analogue of these results in this context.

We should a.lso mention that in the case where '0 E ID>, the formula (19) follows easily from the formulae (16) and (11). For (0 E E2n+2(b), the result is more delicate and the key point of the proof is to show that

(20) 1")«(0) = (I, k~o"h,


b P to . for every function 1 E ll(b) and then show that Tbk~,1 = Cpk'o,1 use once agam (16).

A consequence of (20) and Theorem 9 is that if Co E E2l+2(b), then k~" tends

weakly to k~o,1 as w approaches radially to (0. It is natural to ask if this weak convergence can be replaced by norm convergence. In other words, is it true that Ilk~,1 - k~o,tllb ~ 0 as w tends radially to (o?

In [2], Ahern and Clark claimed that they can prove this result for the case where b is inner and l = O. For general functions b in the unit ball of HOC), Sarason [67, Chapter V] got this norm convergence for the case l = O. In [45], we answer this question in the general case and get the following result.

Theorem 13 (Fricain-Mashreghi, 2008). Let b be a point in the unit ball of HOC), let 1 be a nonnegative integer and let (0 E E21+2(b). Then

IIk~" - k~o,tllb ~ 0, as w tends mdially to (0.

The proof is based on explicit computations of IIk~" \b and k~ I b and we use a nontrivial formula of combinatorics for sums of binomial coefficient. We should mention that we have obtained this formula by hypergeometric series. Let us also mention that Bolotnikov - Kheifets got a similar result in {20] using different techniques and under their condition (15).

We will now discuss the weighted norm inequalities obtained in {16]. The main goal was to get an analogue of Theorem 4 in the setting of the de Branges-Rovnyak spaces. To get these weighted Bernstein type inequalities, we first used a slight modified formula of (19).

Proposition 1 (Baranov-Fricain-Mashreghi, 2009). unit ball of HOC). Let (0 E ]I)) U E 21+2 (b), lEN, and let

Let b be m the

(21) ftP (1'):= b(1' )~~=o (~~~)(-I)JbT((o)b1(() '0,1 ." .,,0 (1 _ (0()1+1 '

(ET.

( b )1+1 2 Then k,o E H and ft~ I E L2(p). Moreover, lor every function I E 1i b), we

have 0,

(22) 1(1)«(0) = l!(l/«()(I(k~o)I+l«()dm«) + 19«)P«)(lft~o'I«)dm«)). where 9 E H2(p) is such that Tbl = Cpg.

We see that if b is inner, then it is clear that the second integral in (19) is zero (because p == 0) and we obtain the formula (2) of Ahern-Clark.

We now introduce the weight involved in our Bernstein-type inequalities. Let 1 < p ::; 2 and let q be its conjugate exponent. Let lEN. Then, for % E closJl)), we define

Wp,I(Z):= min{lI(k!)/+lIl;pl/(p'+1), IIpl/qft~,/II;pl/(p/+1)}j

we assume Wp,I«) = 0, whenever ( E T and at least one of the functions (k~)/+1 or pl/qft~,1 is not in Lq(T).

The choice of the weight is motivated by the representation (22) which shows

that the quantity max{lI(k!)'+l 1l2 , IIp1/2ft~,11I2} is related to the norm of the functional 1 1-+ I(I)(Z) on 1i(b). Moreover, we strongly believe that the norms of


reproducing kernels are an important characteristic of the space 1£(b) which captures many geometric properties of b. Using similar arguments as in the proof of Proposition 1, it is easy to see that p1/qii.'<.1 E Lq(1I') if ( E Eq(I+1) (b). It is also natural to expect that (k~)1+1 E Lq(1I') for (E EQ(I+1)(b). This is true when b is an inner function, by a result of Cohn [28]; for a general function b with q = 2 it was noticed in [16]. However, it seems that the methods of [16,28] do not apply in the general case.

II f E 1£(b) and 1 < P $ 2, then (f(l)w".d(x) is well-defined on 11'. Indeed it follows from [44] that f(l)() and W".I() are finite if (E E21+2(b). On the contrary if ( f/. E21+2(b). then l\(k~)1+11\2 = +00. Hence, l\(k~)1+1I1Q = +00 which, by

definition, implies W".l() = 0, and thus we may assume (f(!)W".I)() = o. In the inner case, we have pet) = 0, then the second term in the definition of

the weight W".l disappears and we reCOver the weights considered in [12]. It should be emphasized that in the general case both terms are essential: in [16] we give an example where the norm IIp1 QJt~ 1 IIQ cannot be majorized uniformly by the norm

• kb 1+1

% Q'

Theorem 14 (Baranov-Fricain-Mashreghi, 2009). Let /L be a Carles on measure on clos JIll , let lEN, let 1 0 such that

23 IIf(l)w".dl£2(I') ~ Cllfllb, f E 1£(b).

If p = 2, then T2.1 is of weak type (2,2) as an operator from 1£(b) into L2 (/L).

The proof of this result is based on the representation (22) which reduces the problem of Bemstein type inequalities to estimates on singular integrals. In particular, we use the following estimates on the weight: for 1 0 such that

(1 - Iz\)1 W".I(Z) ~ A (l-lb(z)\)pl/(Q(pl+l))' Z E 1Dl.

To apply Theorem 14 one should have effective estimates for the weight w".1. that is, for the norms of the reproducing kernels. In the following, we relate the weight wp I to the distances to the level sets of Ibl. We start with some notations. Denote by u.(b) the boundary spectrum of b, Le.,

u.(b):= (E 11': liminflb(z)1 < 1 . z-+( zED

Then closu.(b) = u(b) n 11' where u(b) is the spectrum defined at the begining of this section. For e E (0,1), we put

-neb, e) := {z E JIll : Ib(z)\ < e:} and neb, e) := ui(b) U neb, e:).

Finally, for ( E T, we introduce the following two distances

and d~():= dist(,n(b,e)).


Note that whenever b = I is an inner function, for all , E 00.(1), we have

lim infI1(z)1 = 0, .z-+< .zeD

and thus de«) = de «(), ( E T. However, for an arbitrary function b in the unit ball of Hoc, we have to distinguish between the distance functions de and de.

Using fine estimates on the derivatives W «)\, we got in [16] the following result.

Lemma 1. For each p > 1, 1 ~ 1 and e E (0,1), there exists G = G(e,p,l) > 0 such that

(24)

for all ( E T and 0 ~ r ~ 1.

This lemma combined with Theorem 14 imply immediately the following.

Corollary 1 (Baranov - Fricain - Mashreghi, 2009). For each e E 0, 1 and lEN, there exists G = G(e, I) such that

IIf(l)~112 ~ Gllfllb, / E ll(b).

As we have said in Section 2, weighted Bernstein-type inequalities of the form (23) turned out to be an efficient tool for the study of the so-called Carleson-type embedding theorems for backward shift invariant subspaces Iq. Notably, methods based on the Bernstein-type inequalities allow to give urufied proofs and essentially generalize almost all known results concerning these problems see [12,15] . Here we obtain an embedding theorem for de Branges- Rovnyak spaces. The first statement generalizes Theorem 6 (of Volberg-Treil) and the second statement generalizes a result of Baranov (see [12]).

Theorem 15 (Baranov-Fricain-Mashreghi, 2009). Let p be a positive Borel measure in closlI1>, and let e E (0,1).

(a) Assume that JL(S«(,h)) ~ Kh for all CarLes on squares S("h sahsfymg

Sec, h) n neb, e) 10. Then 1-£(b) C L2 (JL), that is, there is a constant G> 0 such that

Ilfllp(!') ~ Gil/lib, / E ll(b).

(b) Assume that JL is a vanishing Carleson measure for 1l(b), that is,

JL(S«(,h))/h -+ 0 whenever S("h) n n(b,e) 1 0 and h -+ O. Then the embedding 1-£(b) C L2 (JL) is compact.

Note that whenever b = I is an inner function, the sufficient condition that appears in (a) of Theorem 15 is equivalent to the condition of Volberg-Treil theorem because in that case (as already mentionned) we always have 00.(1) C closn(l,e) for everye > O.

In Theorem 15 we need to verify the Carleson condition only on a special subclass of squares. Geometrically this means that when we are far from the spectrum u(b), the measure JL in Theorem 15 can be essentially larger than standard Carleson measures. The reason is that functions in 'H.(b) have much more regularity at the points, E T \ u(b) (see Theorem 8). On the other hand, if Ib«)1 :5 6 < 1, almost everywhere on some arc reT, then the functions in 1l(b) behave on r essentially


25 z E JI)),

In [161. we also discuss another application of our Bernstein type inequalities to the problem of stabIlity of Riesz bases consisting of reproducing kernels in 'H,(b).

4. Weighted backward shift invariant subspaces

Let US now turn to weighted backward shift invariant subspaces. As will be b low, the weighted versions we are interested in appear naturally in the

context of kernels of Toeplitz operators. In Section 4.1 we will present an example sh wing that the generalization of the Ahern - Clark result to this weighted situation is far from being immediate. For this reason we will focus essentially on analytic continuatIOn in thIs section.

For an outer function 9 in HP, we define weighted Hardy spaces in the following w»way:

1 • __ U1>_ .. - fl--

9 'II"

---'II"

Clearly / H /g induces an isometry from HP(\g\P) onto HP. Let now I be any inner function.

We shall discuss the situation when p = 2. There are at least two ways of generalizing the backward shift invariant subspaces to the weighted situation. We first discuss the simple one. AB in the unweighted situation we can consider the orthogonal complement of shift invariant subspaces I H2 (\g\2), the shift 5' lfl 9 2) -+ lfl( g\2) being given as usual by 5/(z) = z/(z). The weighted scalar product is defined by

'II"

Then

(5j,h) 92 = (zjg,hg) = {fg,zhg} = (jg,P+(zhg) = ,

108 E. FRICAIN AND A. HARl'MANN

In other words, with respect to the scalar product (-,.) 92 the adjoint shift is given by S; := ~p+gz, and

K~'9 := (IH2(\gI2)).L = {f E H2(lgI2) : (fg,lhg) = O,h E ~(g 2)}

= {J e H2(lgI2) : (fg,Ih) = 0, hE H2}

= {f e H2(lgI2): (P+(Ifg),h) = O,h E H2}

= {f e H2(lgI2): I!P+ifg,h) =O,h E H2(g2)}. \g 1912

So, K~'9 = ker(1/g)p+ig) = ker(iP+ig). Setting Pf := (I 9 p_ig we get a self-adjoint projection such that

K~'9 = Pf H2(lgI2) = !PI (gH2(lgI2)) = !PIH2 = !.K~, 9 9 9

where PI is the unweighted orthogonal projection onto K'f. Hence, in this situatIon continuation is completely determined by that in K~ and that of 1 g.

We will thus rather consider the second approach. The spaces to be discussed now appear in the context of kernels of Toeplitz operators. Set

KHlgIP) = W(lgIP ) n IH~(lg P ,

where now Hb(lgIP) = zHP(lgIP). The connection with Toeplitz operators arises in the following way: if tp = Ig 9

is a unimodular symbol, then ker Tcp = gKJ(lgI2) (see [48] . Conversely, whenever o f. f e ker Tcp, where tp is unimodular and / = J 9 is the inner-outer factorization of /, then there exists an inner function 1 such that I{J = 19 9 see also [48] .

Note also that the following simple example shows that in general K:,g icI different from KJ(lgI 2). Let l(z) = z be the simplest Blaschke factor. Then H2(lgI2) nIHg(lgI2) = H2(lgI2) nH2(lgl) = C whenever 9 IS rigid more on rigidity follows later). On the other hand, (1/g)KJ is the one-dimensional space spanned by 1/g which is different from C when 9 is not a constant.

The representation kerTcp = gKHlgIP) is particularly interesting when 9 is the extremal function of kerTcp. Then we know from a result by Hitt [50] see also [66] for a de Branges Rovnyak spaces approach to Hitt's result) that when p = 2, kerTcp = gKJ, and that 9 is an isometric divisor on kerTtp = gKJ (or 9 is an isometric multiplier on K1). In this situation we thus have Kl( g12) = KJ. Note, that for p f. 2, if 9 is extremal for gKf(lgIP), then KfOg P) can still be imbedded into KJ when p > 2 and in Kf when p E (1,2) (see [48], where it is also shown that these imbeddings can be strict). In these situations when considering questions concerning pseudocontinuation and analytic continuation, we can carry over to Kf(lglP) everything we know about KJ or Kf, i.e., Theorems 1 and 7. Concerning the Ahern Clark and Cohn results however, when p f. 2, we lose information since condition (ii) in Theorem 2 depends on p.

In general the extremal function is not easily detectable (explicit examples of extremal functions were given in [48]), in that we cannot determine it, or for a given 9 it is not a simple matter to check whether it is extremal or not. So a natural question is to know under which conditions on 9 and I, we can still say something about analytic continuation of functions in Kf(lgIP). It turns out that Moeller's result is valid under an additional local integrability condition of 1/g on a closed


arc not meeting the spectrum of I. Concerning the regularity questions in point8 contained in the spectrum, the situation is more intricate. As mentioned earlier, an example in this direction will be discussed at the end of this section.

Regularity of functions in kernels of Toeplitz operators have been considered by Dyakonov. He in particular establishes global regularity properties of functions in the kernel of a Toeplitz operator such as being in certain Sobolov and Besov spaces [35] or Lipschitz and Zygmund spaces [41] depending on the smoothness of the corresponding Toeplitz operator.

The following simple example hints at some difference between this situation and the unweighted situation or the context of de Branges-Rovnyak spaces discussed before. Let [ be arbitrary with -1 ¢ 0'(1), and let g(z) = 1 + z, so that 0'(1) is far from the only point where 9 vanishes. We know that kerTl9Ig = gKHlgIP). We first observe that (1 + z)/(l + z) = z. Hence,

1 E K~l = kerT,,1 = kerTl9Ig = gKf(igIP)

So, KHlg P) contains the function l/g which is badly behaved in -1, and thus ('Annot extend analytic.ally through -l.

This observation C8.n be made more generally as stated in the following result [46].

Proposition 2 (Hartmann 2008). Let 9 be an outer function in HP. If kerTg 9 =F {O} contains an inner junction, then l/g E KHlgIP) for every inner functwn [.

Note that if the inner function J is in kerTgIg then T".Jgjg1 = 0, and hence 1 E kerT-.,g 9 = gKj(lgl2) and l/g E Kj(lgI2), which shows that with this simple argument the proposition holds with the more restrictive condition 1 = J.

Let us comment on the case p = 2: The cla.im that the kernel of TgIg contajns an inner function implies in particular

that Tg 9 is not injective 8.nd so g2 is not rigid in HI (see [67, X-2]), which means that it is not uniquely determined up to iii. real multiple by its argument (or equivalently, its normalized version g2/lIg2lh is not exposed in the unit ball of HI).

It is clear that if the kernel of a Toeplitz operator is not reduced to {O}or equivalently (since p = 2) g2 is not rigid then it contains an outer function Gust divide out the inner factor of any nonzero function contained in the kernel). However, Toeplitz operators with nontrivial kernels containing no inner functions CAn be easily constructed. Take for instance Tzgolgo = TzTgoI go ' where go(z) = 1- z)a and Q E (0, ~). The Toeplitz operator Tgolgo is invertible (lgol2 satisfies

the MUckenhoupt (A2 ) condition) and (Tilolgo)-l = gOP+/o [61] so that the kernel of Tzg 9 is given by the preimage under Tgolgo of the constants (which define the kernel of Tz). Since gOP+(c/go) = cgo/go(O), e being any complex number, we have kerTzg Igo = ICgo which does not contain any inner function.

So, without any condition on g, we cannot hope for reasonable results. In the above example, when p = 2, then the function g2(Z) = (1 + z? is in fact not rigid (for instance the argument of (1 + z)2 is the same as that of z). AB already pointed out, rigidity of g2 is also characterized by the fact that Tgl9 is injective (see [67, X-21). Here Tgl g = Ti the kernel of Which is C. From this it can also be deduced that g2 is rigid if and only if HP(lgIP) n HP(\g\P) = {O} which indicates again that


rigidity should be a.ssumed if we want to have K}'(\g 1') reasonably defined. (See [53] for some discussions on the intersection HP(\g\P) n HP( 9 1').)

A stronger condition than rigidity (at least when p = 2) is that of a Muckenhoupt weight. Let us recall the Muckenhoupt (Ap) condition: for general 1 < p < 00

a weight w satisfies the (AI') condition if

B := I 8U~~~ of T { \~\1 w(x) dx x C~\l W-1/(P-l) (x) dx Y-1

} < 00.

When p = 2, it is known that this condition is equivalent to the S<H:a1led HelsonSzego condition. The Muckenhoupt condition will play some role in the results to come. However, our main theorem on analytic continuation Theorem 11) works under a weaker local integrability condition.

Another observation can be made now. We have already mentioned that rigidity of y2 in HI is equivalent to injectivity of Tg / g , when 9 is outer. It is also clear that Tg/g is always injective so that when g2 is rigid, the operator Tg 9 is injective with dense range. On the other hand, by a result of Devinatz and Widom see, e.g., [59, Theorem B4.3.1]), the invertibility of Tg / g , where 9 is outer, is equivalent to \g\2 being (A2). So the difference between rigidity and (A2 is the surjectivity in fact the closedness of the range) of the corresponding Toeplitz operator. A criterion for surjectivity of noninjective Toeplitz operators can be found in [47}. It appeals to a parametrization which was earlier used by Hayashi [49) to characterize kernels of Toeplitz operators among general nearly invariant subspaces. Rigid functions do appear in the characterization of Hayashi.

As a consequence of Theorem 17 below analytic continuation can be expected on arcs not meeting the spectrum of I when IglP is (AI') see Remark 1 . However the (AI') condition cannot be expected to be necessary since it is a global condition whereas continuation depends on the local behaviour of I and g. We will even give an example of a nonrigid function 9 (hence not satisfying the Ap condition for which analytic continuation is always possible in certain points of T where 9 vanishes essentially.

Closely connected with the continuation problem in backward shift invariant subspaces is the spectrum of the backward shift operator on the space under consideration. The following result follows from [9, Theorem 1.9): Let B be the backward shift on HP(\g\P), defined by BI(z) = (J - 1(0)) z. Clearly, K}'( 9 1') is invariant with respect to B whenever I is inner. Then, u(B I Kf( 9 1')) = uap(B Kf( 9 1'», where uap(T) = {A E C : 3(fn)n with II In II = 1 and (A - T)/n -+ O} denotes the approximate point spectrum of T, and this spectrum is equal to

1l' \ {I/( E 1l' : every I E Kf(lgIP) extends analytically in a neighbourhood of (}.

The aim is to link this set and utI). Here we will need the Muckenhoupt condition. Then, as in the unweighted situation, the approximate spectrum of B I Kf(lgIP) on 1l' contains the conjugated spectrum of I. We will see later that the inclusion in the following proposition [46] actually is an equality.

Proposition 3 (Hartmann, 2008). Let 9 be outer in HI' such that IglP is a Muckenhoupt (Ap)-weight. Let I be an inner function with spectrum u(I). Then

utI) C uap(B I Kf(lg\p)).

We now come to the main result in the weighted situation (see [46]).


Theorem 17 (Hartmann, 2008). Let 9 be an outer function in H", 1 q, l/p + l/q = 1, with l/g E LB(r), then every function f E Kf(lgl") extends analytically through r if and only ifr does not meet 0"(1).

Note that in [46] only the sufficiency part of the above equivalence was shown. However the condition that r must not meet 0"(1) is also necessary (even under the a priori weaker condition of continuation through r) as follows from the proof of Theorem 18 below. A stronger version of Theorem 17 can be deduced from [5, Corollay 1 of Theorem 3]

It turns also out that like in the de Branges Rovnyak situation discussed in Theorem 8 for analytic continuation it is actually sufficient to have continuation. This result is new, and we will state it as a. theorem provided with a. proof. It is

on idea.s closed to the proof of the previous theorem.

Theorem 18. Let 9 be an outer function in HP, 1 < p < 00 and I an inner funcnon with associated spectrum 0"(1). Let r be an open arc in 'll'. Suppose that every functwn f E Kf(lgIP) extends continuously to r then rnO"(I) = 0, and every funcnon In Kf( glP) extends analytically through f.

PROOF. Observe first that obviously k{ E K;(lgI 2). By the Schwarz reflection

principle, in order that k{ continues through f we need that f does not meet 0"(1) note that closf could meet 0"(1)).

As in the unweighted situation, every meromorphic function J / I, J = I"iji E -- - ~

K; 9 2 ,admits a pseudo continuation 'IjJ, defined by 'IjJ(z) = E .. ~o 'IjJ(n) 1/ z .. in the ~

exterior disk De = C \ clos 1Dl. Fix f any closed subarc of f. Since 0"(1) is closed, the distance between 0"(1)

and fo is strictly positive. Then there is a neigbourhood of fo intersected with IDl where I z) ~ 5> O. It is clear that in this neighbourhood we are far away from the part of the spectmm of I contained in 1Dl. Thus I extends analytically through fo. For what follows we will call the endpoints of this arc (1 := eit1 and (2 := eit2

oriented in the positive sense). The following argument is in the spirit of Moeller [56] and based on Morera's

theorem. Let us introduce some notation (see Figure 1). For suitable ro E (0,1) let no = {z = reit E 1Dl: t E [tbt2],ro ~ r < I}. and

flo = {z = eit /r E IDle: t E [tl' t2], ro ~ r < I}. Define

F(z) = J(z)/I(z) -'IjJ(z)

z E no; -z E no. - -

By construction this function is analytic on no u no and continuous on no Uno. -Such a function is analytical on no uno. 0

Remark 1. It is known (see, e.g., [571) that when IglP E (Ap), 1 ro. Take r E (ro,p)· Then in particular l/g E U, where l/r+l/s = 1. Since r q which allows to conclude that in this situation 1/ 9 E U (r) for every r c 'll' (s independant of r).

We promised earlier an example of a nonrigid function 9 for which analytic continuation of iG-functioDS is possible in certain points where 9 vanishes.


FIGURE 1. The regions no and no

Example. For a E (0, ~), let g(z) = (1 + z)(l - z)a. Clearly 9 is an outer function vanishing essentially in 1 and -1. Set h (z) = z (1 - Z 2a, then by similar arguments as those employed in the introducing example to this section one can check that arg g2 = arg h a.e. on 1l'. Hence 9 is not rigid it is the "big" zero in -1 which is responsible for nonrigidity). On the other hand, the zero in +1 is "small" in the sense that 9 satisfies the local integrability condition in a neighbourhood of 1 as required in the theorem, so that whenever 1 has its spectrum far from 1, then every K;(lgI 2 )-function can be analytically continued through suitable arcs around 1.

This example can be pushed a little bit further. In the spirit of Proposition 2 we check that (even) when the spectrum of an inner function 1 does not meet -1, there are functions in Kf( \g II') that are badly behaved in -1. Let again go (z = 1 - z a.

Then g(z) (1 + z)(l- z)a _go(z) --= =Z--. g(z) (1 + z)(l - z)a 90(Z)

As already explained, for every inner function 1, we have kerT[g 9 = gKf( 9 1'), so that we are interested in the kernel kerT[glg' We have Tig gl = 0 when 1 = Iu and'U E kerTglg = kerT"zgo/go = ego (see the discussion just before the proof of Proposition 2). Hence the function defined by

F(z) = I(z) = I(z)go(z) = I(z) g(z) g(z) 1 + z

is in Kf(lgIP) and it is badly behaved in -1 when the spectrum of 1 does not meet -1 (but not only).

The preceding discussions motivate the following question: does rigidity of 9 suffice to get analytic continuation for Kf(lgI2 )-function whenever 0'(1) is far from zeros of g?

Theorem 17 together with Proposition 3 and Remark 1 allow us to obtain the following result. We should mention that it is easy to check that HP(lgl") satisfies


the conditions required of a Banach space of analytic functions in order to apply the results of [9}.

Corollary 2 (Hartmann, 2008). Let 0 be outer in HP such that 10lP is a Muckenhoupt (Ap) weight. Let I be an mner function with spectrum 0'(1) = {A E closl!ll: liminf,,-+-x I(z) = o}. Then 0'(1) = O' .. p(B I Kf(loIP».

Another simple consequence of Theorem 17 concerns embeddings. Contrarily to the situations discnssed in Sections 2 and 3, the weight is here on the Kf-side.

Corollary 3 (Ha .. tmann, 2008). Let I be an inner function with spectrum 0' I). If reT is a closed arc not meeting 0'(1) and if 9 is an outer function in HP such that 9 ~ 0 on 11' \ r for some constant 0 > 0 and I/O E LB(r), s > q, 1 p + 1 q = 1. Then Kf (loIP) c Kf. If moreover 0 is bounded, then the last mcluswn IS an equality.

Suppose now p = 2. We sha.ll use this corollary to construct an example where Kl 9 2) = Kl without 0 being extremal for gKJ(lgI2). Recall from Hitt's result [50], that when 9 is the extremal function of a nearly invariant subspace M c H2, then there e..xists aD inner function I such that M = gKJ, and 9 is an isometric multiplier on Kl so that KJ = KJ (lgI2). Recall from [48, Lemma 3] that a function 9 is extremal for gKJ(lgI2) if IIIgl2dm = 1(0) for every function I E KJ(lgI2). Our example is constructed in the spirit of [48, p. 356]. Fix Q E (0, ~). Let 'Y z = 1- z)O< and let 9 be an outer function in H2 such that Igl2 = Re'Y a.e. on T such a function clearly exists). Let now I = BA be an infinite Blaschke product with 0 E A. If A accumulates to points outside 1, then the corollary shows that K1 = K1 9 2). Let us check that 9 is not extremal. To this end we compute J k 9 2dm for A E A (recall that for A E A, k).. E KJ = KJ(lgI2»:

26 1

k"g\2dm= k"Re'Ydm=2 k"'Ydm+ k"'Ydm

1 1 1 = 2 k).. (O)-y (0) + 2(k",'Y) = 2(1 + (1- A)a)

which is different from k,,(O) = 1 (except when A = 0). Hence 9 is not extremal. We could also have obtained the nonextremality of 9 from Sarason's result

[64, Theorem 21 using the parametrization 9 == a/(l - b) appearing in Sarason's and Hayashi's work [46] for details on this second argument).

It is clear that the corollary is still valid when r is replaced by a finite union of intervals. However, we can construct an infinite union of intervals r = Un~l r n

each of which does not meet 0'(1), an outer function 9 satisfying the yet weaker integrability condition l/g E U(r), 8 < 2, and Igl ~ 0 on 11' \ r, and an inner function I such that KJ(lgI2 ) rt Kr The function 9 obtained in this construction does not satisfy 9 2 E (A2 ). (See [46] for details.)

Another simple observation concerning the local integrability condition l/g E L" r), 8 > q: if it is replaced by the global condition 1/ 9 E U(11'), then by Holder's inequality we have an embedding into a bigger backward shift invariant subspace:

Proposition 4 (Hartmann, 2008). Let 1 q such that 1/9 E £0(11'), then for r with l/r = l/p + l/s we have 0'( 9 P) C Lr.


So in this situation we of course also have K}'(\gIP) C KJ. In particular, every function f e Kr(lgIP) admits a pseudocontinuation and extends analytically outside 0'(1). Again the Ahern - Clark condition does not give complete information for the points located in the spectrum of I since (ii) of Theorem 2 depends on p.

When one allows 9 to vanish in points contained in 0'(1), then it is possible to construct examples with IglP e (Ap) and K}'(lgIP) ¢. K}': take for instance 1= BA the Blasche product vanishing exactly in A = {1-1/2n}n and g(z) = (l-z)O, where 0: e (0, l) and p = 2 (see [46] for details; the condition Igl2 e (A2 ) is required in the proof to show that K;(lgI2) = P+ ((1/y)KJ) -see Lemma 2 below-which gives an explicit description of K; in terms of coefficients with respect to an unconditional basis). The following crucial example is in the spirit of this observation.

4.1. An example. In the spirit of the example given in [46, Proposition 4) we shall now discuss the condition (ii) of Theorem 2 in the context of weighted backward shift invariant subspaces.

We first have to recall Lemma 1 from [46]:

Lemma 2 (Hartmann, 2008). Suppose IglP is an (Ap) weight and I an inner function. Then Ao = P+1/y: HP -+ HP(lgIP) is an isomorphism of K1 onto KHI9IP ). Also, for every >. e lIJ) we have

(27) A k _ k),,(J.L) o )" - g(>.)'

We return to the situation p = 2. Take g(z) = (1- z)O with 0: E (0,1 2 . Then 1912 is (A2 ). Let

rn = 1- ;n' On = (1- rn)S = 2:8 ' An = Tnei9 .. ,

where s e (0, ~). Hence the sequence A = {An}n tends tangentially to 1. Set 1= BA • We check the Ahern-Clark condition in (= 1 for 1= 0 (Which means that we are just interested in the existence of nontangentiallimits in ( = 1). Observe that for B e (0, ~) we have

(28) 11 r ei9n 12 '" (1 r)2 + 02 _ 1 + 1 ,.",. 1 - n - - n n - 22n 22n8 - 22ns '

and so when q > 1

(29) " 1- r~ "''' 1/2n

,.",." 2n (8q-l) LJ 11 - rnei9n Iq - LJ 1/2n8q - LJ . n~l n~l n~l

The latter sum is bounded when q = 2 which implies in the unweighted situation that every function in the backward shift invariant subspace K~ has a nontangential limit at 1. Note also that since Igl2 e (A2 ), by Proposition 4 and comments thereafter, K;(\gI2) imbeds into some KJ, r < 2. Now taking q = r' > 2, where 1/r + 1/r' = 1, we see that the sum in (29) diverges when sr' ~ 1 and converges for sT' < 1. So depending on the parameters s and 0: we can assert continuation or not. It will be clear a. posteriori that in our situation r has to be such that Br' ~ 1.

Note that 0'(1) n'll' = {l}, which corresponds to the point where 9 vanishes. Clearly, A is an interpolating sequence, and so the sequence {k)"n I II k)"n 1\2}n is a normalized unconditional basis in Kr2. This means that we can write Kl 12 ( ~ )

II II: "n III

REGULARITY ON THE BOUNDAl\Y ltr.

weaning that f E K~ if and only if

I L k>. .. = n>1 an Ilk>. .. 112

with Ln>l lanl2 < 00 (the last sum defines the square of an equivalent noriU in K~). -As already mentioned 1912 is Muckenhoupt (A2). This implies in particular

that we have the local integrability condition 1/9 E L'(r) for some 8 > 2 and ran arc containing the point 1. Moreover, we get from (27)

, "

and {k>'n (g(>"n)\\k>'nl\2n" is an unconditional basis in KHIgI2) (almost normalized in the sense that I\Ao(k>. .. /\Ik>. .. 112)llIg\2 is comparable to a constant independant of n). Hence for every a = (an)n with Ln>llo~1 < 00, we have -

an k>... 2 2

,,>1 9 n >' .. 2 -To fix the ideas we will now pick 0" = 1/n1 / 2+s for some e > 0 so that

L:nank>... k>'nlh is in K~, and hence 10 E KHIgI2). Let us show that 10 does not have a. nont,a.ngentiallimit in 1. Fix t E (0,1). Then

t = On k>." (t)

-

So 2n (.o-1/2) 1/2ns

R g(>..11.) Ilk>. .. 112 R~lop;2(1/(1-t))

"" (1- t)2. 2"1R - (1- t)2. 2"1 loP;2(1/(1-t)) R~1op;2(1/(1-t))

~ (1 - tp-2.,


where "y = 8 + ! - sa + 6 for an arbitrarily small 6 (this compensates the term n1/ 2H). So "y - 28 == ! - s(1 + a) + 6 which can be made negative by choosing 8

closely enough to !. We conclude that the function f 0. is not bounded in 1 and thus cannot have a

nontangential limit in , = 1

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Paris, 1967. 72. A. L. Vol berg, Thin and thick familIes of rational fracnona, Complex Analysis and Spectral

Theory (Leningrad, 1979/1980), Lecture Notes in Math., vol. 864, Springer, Berlin, 1981, pp.440 480.

73. A. L. Volberg and S. R. Treil, Embedding theorems for ,nvanant subspaces of the Inver"e sh'ft operatar, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Stek!ov. (LOMI) 149 (1986),

~t"d L nt'

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REGULAIUTY ON THE ROUNDARY 119

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Cent ... d. Reehercbee MatMmatiqu,," CRM Proceedings and I ecture Not .. Volume 51, lOlO

The Search for Singularities of Solutions to the Dirichlet Problem: Recent Developments

Dmitry Khavinson and Erik Lundberg

ABSTRACT. This is a survey article based on an invited talk delivered by the first author at the CRM workshop on Hilbert Spaces of Analytic Functions held at CRM, Universite de Montreal, December 8-12,2008.

1. The main question

Let 0 be a smoothly bounded domain in Rn. Consider the Dirichlet problem DP) in 0 of finding the function u, say, E C2(n) n C(n) and satisfying

1.1 Au = 0

ulr =v ,

where A = E;=l a2 /ax~ and r := an, v E c(r). It is well known since the early 20th century from works of Poincare, C. Neumann, Hilbert, and Fredholm that the solution u exists and is unique. Also, since u is harmonic in n, hence leal-analytic there, no singularities can appear in O. Moreover, assuming r := an to consist of real-analytic hypersuIfaces, the more recent and difficult results on "elliptic regularity" assure us that if the data v is real-analytic in a neighborhood ofn then u extends as a real-analytic function across an into an open neighborhood 0' ofO. In two dimensions, this can be done using the reflection principle. In higher dimensions, the boundary can be biholomorphically "flattened," but this leads to a general elliptic operator for which the reflection principle does not apply. Instead, analyticity must be shown by directly verifying convergence of the power series representing the solution through difficult estimates on the derivatives (see [14]).

Question. Suppose the data 11 is a restriction to r of a ''very good" function, sayan entire function of variables Xl, X2, • •• , X n • In other words, the data presents no reasons whatsoever for the solution u of (1.1) to develop singularities.

(i) Can we then assert that all solutions u of (1.1) with entire data vex) are also entire?

2000 Malhematt.C$ SubJect Cla8Bificabon. Primary 31B20; Secondary 30H20. Both authors gratefully acknowledge partial 8Upport from the National Science Foundation. This is the final fulW of the paper.

(£)2010 American Mathematical Socl • .,

121

122 D. KHAVINSON AND E. LUNDBERG

(ii) If singularities do occur, they must be caused by geometry of r interacting with the differential operator t::,.. Can we then find data Vo that would force the worst possible scenario to occur? More precisely, for any entire data v, the set of possible singularities of the solution 1£ of (1.1) is a subset of the singularity set of 1£0, the solution of (1.1) with data Vo.

2. The Cauchy problem

An inspiration to this program launched by H. S. Shapiro and the first author in [22] comes from reasonable success with a similar program in the mid 1980's regarding the analytic Cauchy problem (CP) for elliptic operators, in particular, the Laplace operator. For the latter, we are seeking a function 1£ with t::,.u = 0 near r and satisfying the initial conditions

(2.1) {(u - v)\r = 0 V(u - v)\r = 0

where v is assumed to be real-analytic in a neighborhood of r. Suppose as be{ re that the data v is a "good" function (e.g., a polynomial or an entire function. In that context, the techniques developed by J. Leray [26] in the 19505 and j intly with L. Gar-ding and T. Kotake [15]) together with the works of P. Ebenfelt [11, G. Johnsson [18], and, independently, by B. Sternin and V. Shatalov [33 in Russia and their school produced a more or less satisfactory understandIng of the situatIOn. To mention briefly, the answer (for the CP) to question 1 in two dimensi DS is essentially "never" unless r is a line while for (ii) the data mining all pOSSlble singularities of solutions to the CP with entire data 18 v = X 2 = L r see [19-21,34] and references therein).

3. The Dirichlet problem: When does entire data imply entire solution?

Let us raise question (i) again for the Dirichlet problem: Does real entire data v imply entire solution u of (1.1)?

In this section and the next, Jl» will denote the space of polynomials and PH the space of polynomials of degree ~ N. The following pretty fact goes back to the 19th century and can be associated with the names of E. Heine, G. Lame, M. Ferrers, and probably many others (cf. [20]). The proof is from [22] (cf. [2,3]).

Proposition 3.1. If 0. := {x : LX~ a~ - 1 < O,al > ... > a,. > O} is an ellipsoid, then any DP with a polynomial data of degree N has a polynomial solution of degree ~ N.

PROOF. Let q(x) = E x~ / a~ - 1 be the defining function for r := an. The (linear) map T: P -+ ~(qP) sends the finite-dimensional space PH into itself. Tis injective (by the maximum principle) and, therefore, surjective. Hence, for any P, degP ~ 2 we can find Po, degPo ~ degP - 2. TPo = ~(qPo) = ~P. u = P- qPO is then the desired solution. 0

The following result was proved in [22].

Theorem 3.2. Any solution to DP (1.1) in an ellipsoid n with entire data ss also entire.

SEARCH FOR SINGULARITIES OF SOLUTIONS TO THE DIRICHLET PROBLEM 123

Later on, O. Armitage sharpened the result by showing that the order and the type of the data are carried over, more or less, to the solution [1]. The following conjecture has also been formulated in [22].

Conjecture 3.S. Ellipsoids are the only bounded domains in ]Rn for which Theorem 3.2 holds, i.e., ellipsoids are the only domains in which entire data implies enure solution for the DP (1.1).

In 2005 H. Render [30] proved this conjecture for all algebraically bounded domains n defined as bounded components of {q,( x) < 0, q, E IP N} such that {q,(x) = O} is & bounded set in ]Rn or, equivalently, the senior homogeneous part q,N(X) of q, is elliptic, i.e., Iq,N(X)1 ~ Clxl N for some constant C. For n = 2, an ('asier version of this result was settled in 2001 by M. Chamberland and O. Siegel [6]. At the beginning of the next section we will outline their argument, which establishes similar results as Render's for the following modified conjecture.

Conjectm e 3.4. Ellipsoids are the only surfaces for which polynomial data unpiles polynomtal solution.

Remark. We will return to Render's theorem below. For now let us note that, unfortunately, it already tells us nothing even in 2 dimensions for many perturba.tions of a unit disk, e.g., n := {x E ]R2 : x2 + y2 -1 + eh(x, y) < O} where, say, h is & harmonic polynomial of degree> 2.

4. When does polynomial data imply polynomial solution?

Let'Y = {q,(x) = O} be a bounded, irreducible algebraic curve in ]R2. If the OP posed on 'Y has polynomial solution whenever the data is a polynomial, then as Cbamberla.nd a.nd Siegel observed, (a) 'Y is an ellipse or (b) there exists data f E IP such that the solution u E lP of OP has degu > degf.

In case (b) u - fl,. = 0 implies that q, divides u - f by Hilbert's Nullstelensatz, and, since degu = M > degf, UM = q,k9! where q,k and UM are the senior homogeneous ter illS of q, and u respectively. The senior term of u must have the form 'UM = azM + bzM since UM is harmonic. Hence, UM factors into linear factors and so must q,k. Hence 'Y is unbounded. This gives the following result [6].

Theorem 4.1. Suppose deg q, > 2 and q, is square-free. If the Dirichlet problem posed on {q, = O} has a polynomial solution for each polynomial data, then the semor part of q" which we denote by q,N, of order N, factors into real linear terms, namely,

n

q,N = II (ajx - bjy), ;=0

whe,e a" b, are some real constants and the angles between the lines ajx - bjy = 0, for all j, are rational multiples of 11".

This theorem settles Conjecture 3.4 for bounded domains n ~ {q,(x) < O} such that the set {q,(x) = O} is bounded in ]R2. However, the theorem leaves open simple

such as x2 + y2 -1 + e(X3 - 3xy2).

Example. The curve y(y - x){y + x) - x = 0 (see Figure 1) satisfies the condition imposed by the theorem. Moreover, any quadratic data can be

matched on it by a hal monic polynomial. For instance, u = xy(y2 - x2 ) solves the

124 D. KHAVlNSON AND E. LUNDBERG

FIGURE 1. A cubic on which any quadratic data can be matched by a harmonic polynomial.

interpolation problem (it is misleading to say "Dirichlet" pr blem, since there is no bounded component) with data v(x,y) = x2 • On the 0 her hand, one can sh w (nontrivially) that the data x3 does not have polynomial solution.

5. Dirichlet's problem and orthogonal polynomials

Most recently, N. Stylianopoulos and the first author showed that if for a polynomial data there always exists a polynomial solution of the DP 1.1, with an additional constraint on the degree of the solution in terms of the degree of the data (see below), then n is an ellipse [23]. This result draws on the 2007 paper of M. Putinar and N. Stylianopoulos [29] that found a simple but surprising connection between Conjecture 3.4 in JR2 and (Bergman) orthogonal polynomials, i.e. polynomials orthogonal with respect to the inner product (p, q 0 := 10 pi[ dA, where dA is the area measure. To understand this connection let us consider the following properties:

(1) There exists k such that for a polynomial data of degree n there always exists a polynomial solution of the DP (1.1) posed on n of degree ~ n + k.

(2) There exists N such that for all m, n, the solution of (1.1) with data Z"'z" is a harmonic polynomial of degree ~ (N -l)m+n in z and of degree ~ (N-l)n+m in %.

(3) There exists N such that orthogonal polynomials {Pn} of degree n on n satisfy a (finite) (N + I)-recurrence relation, Le.,

where an - 3•n are constants depending on n.


(4) The Bergman orthogonal polynomials of 0 satisfy a finite-term recurrence relation, i.e., for every fixed k > 0, there exists an N(k) > 0, such that alc,R = (ZPn,Pk) = 0, n ~ N(k).

(5) Conjecture 3.4 holds for O.

Putinar and Stylianopoulos noticed that with the additional minor assumption that polynomials are dense in L!(O), properties (4) and (5) are equivalent. Thus, they obtained as a corollary (by way of Theorem 4.1 from the previous section) that the only bounded algebraic sets satisfying property (4) are ellipses. We also have (1) ). (2), (2).: > (3), and (3) ). (4). Stylianopoulos and the first author used the equivalence of properties (2) and (3) to prove the following theorem which has an immediate corollary.

Theorem 5.1. Suppose 130 is C2-smooth, and orthogonal polynomials on n satisfy a (finite) (N + l)-recum:nce relation, in other words property (3) is satisfied. Then, N = 2 and n is an ellipse.

COl'Ollary 5.2. Suppose 130 is a C2-smooth domain for which there exists N such that/or all m, n, the solution of (1.1) with data Z"'ZR is a ha11nonic polynomial 0/ deglw $; (N - l)m + n in z and of degree ~ (N - l)n + m in z. Then N = 2 and 0 IS an elltpse.

SKETCH OF PROOF. First, one notes that all the coefficients in the recurrence relation are bounded. Divide both sides of the recurrence relation above by Pn and take the limit of a,n appropriate subsequence as n -+ 00. Known results on 8S)'lIIptotics of orthogonal polynomials (see [35]) give limn~"" Pn+I/Pn = cJ?(z) on compact subsets of C \ fi, where cJ?(z) is the conformal map of the exterior of 0 to the exterior of the unit disc. This leads to a finite Laurent expansion at 00 for \II w = cJ?-1 (w). Thus, \II (w) is a rational function, so n := C \ n is an unbounded quadrature doma,in, and the Schwarz function (cf. [7,37]) of 130, S(z) (= z on 130) -has a meromorphic extension to O. Suppose, for the sake of brevity and to fix the

M -ideas, for example, that S(z) = czd + L:j=l Cj/(z - Zj) + f(z), where f E H""(O), -and zJ E n. Since our hypothesis is equivalent to 0 satisfying property (2) discussed

above, the data zP(z) = Zn~=l(Z - Zj) has polynomial solution, g(z) + h(z) to

the DP. On r we can replace z with S(z). Write h(z) = h#(z), where h# is a polynomial whose coefficients are complex conjugates of their counterparts in h. We have on r 5.1 S(z)P(z) = g(z) + h#(S(z)),

which is actually true off r since both sides of the equation are analytic. Near Zj, the left-hand side of this equation tends to a finite limit (since S(z)P(z) is analytic -in n oo!) while the right-hand side tends to 00 unless the coefficient Cj is zero. Thus,

5.2) S(z) = czd + f(z).

Using property (2) again with data \z12 = zz we can infer that d = 1. Hence, n is a null quadrature domain. Sakai's theorem [32] implies now that 0 is an ellipse. 0

Remark. It is well-known that families of orthogonal polynomials on the line satisfy a 3-term recurrence relation. P. Duren in 1965 [8] already noted that in ( the only domains with real-analytic boundaries in which polynomials orthogonal


with respect to arc-length on the boundary satisfy 3-term recurrence relations are ellipses. L. Lempert [25] constructed peculiar examples of Coo nonalgebraic Jordan domains in which no finite recurrence relation for Bergman polynomials holds. Theorem 5.1 shows that actually this is true for all C2-smooth domains except ellipses.

6. Looking for singularities of the solutions to the Dirichlet problem

Once again, inspired by known results in the similar quest for solutions to the Cauchy problem, one could expect, e.g., that the solutions to the DP (1.1) exhibit behavior similar to those of the CP (2.1). In particular, it seemed natural to suggest that the singularities of the solutions to the DP outside n are somehow associated with the singularities of the Schwarz potential (function) of an which does indeed completely determine 8n (cf. [21,37]). It turned out that singularities of solutions of the DP are way more complicated than those of the CP. Already in 1992 in his thesis, P. Ebenfelt showed [9] that the solution of the following "innocent" DP in n := {x4 + y4 -1 < a} (the "TV-screen")

(6.1) {~u=a ulan = x2 + y2

has an infinite discrete set of singularities (of course, symmetric with respect to 90" rotation) sitting on the coordinate axes and running to 00 (see Figure 2 .

To see the difference between analytic continuation of solutions to CP and DP, note that for the former

8u (6.2) 8)r:=an = vzCz, z) = vz(z, S(z)),

and since 8u/8z is analytic, (6.2) allows U z to be continued everywhere together with v and S(z), the Schwarz function of 8n. For the DP we have on r

(6.3) u(z, z) = v(z, z)

for u = f + 9 where f and 9 are analytic in n. Hence, (6.3) becomes

(6.4) fez) + g(S(z)) = v(z, S(z)).

Now, v(z, S(z)) does indeed (for entire v) extend to any domain free of singularities of S(z), but (6.4), even when v is real-valued so that 9 = j, presents a very nontrivial functional equation supported by a rather mysterious piece of information that f is analytic in n. (6.4) however gives an insight as to how to capture the DP-solution's singularities by considering the DP as part of a Goursat problem in e 2 (or en in general). The latter Goursat problem can be posed as follows (cf. [36]).

Given a complex-analytic variety f in en, (f n]Rn = r := 8n), find u: '£.;=182u/8z; = a near f (and also in n C ]Rn) so that ulr = tI, where tI is,

say, an entire function of n complex variables. Thus, if f := {¢(z) = a}, where tP is, say, an irreducible polynomial, we can, e.g., ponder the following extension of Conjecture 3.3:

Question. For which polynomials tP can every entire function v be split (Fi&eber decomposition) as v = u + tPh, where ~u = a and u, h are entire functions (cf. [13,36))7


•

, I

• I • • ) • •

•

..

FIGURE 2. A plot ofthe "TV screen" {x4+y4 = 1} along with the first eight singularities (plotted as circles) encountered by analytic continuation of the solution to DP (6.1).

7. Render's breakthrough

Trying to establish Conjecture 3.3 H. Render [30j has made the following ingenious step. He introduced the real version of the Fischer space norm

7.1

where I and 9 are polynomials. Originally, the Fischer norm (introduced by E. Fischer [13]) requires the integration to be carried over all of en and has the property that multiplication by monomials is adjoint to differentiation with the corresponding multi-index (e.g., multiplication by (Ei=l x~) is adjoint to the differential operator A). This property is only partially preserved for the real Fischer norm. More precisely [30j,

7.2) (AI, g) = (f, Ag) + 2 (deg(f) - deg(g)) (f, g)

for homogeneous I, g. Suppose u solves the DP with data \x\2 on 8n ~ {P :::: 0 : deg(P) = 2k, k > 1}.

Then u - X\2 = Pq for analytic q, and thus AIc(Pq) = O. Using (7.2), this (nontrivially) implies that the real Fischer product «(pq)m+21c, qm) between all homogeneous parts of degree m + 2k and m of Pq and q, respectively, is zero. By a tour de force argument, Render used this along with an added assumption on the


senior term of P (see below) to obtain estimates from below for the decay of the norms of homogeneous parts of q. This, in tum yields an if-and-only-if criterion for convergence in the real ball of radius R of the series for the solution u = L~ u."., Urn homogeneous of degree m. Let us state Render's main theorem.

Theorem 7.1. Let P be an irreducible polynomial of degree 2k, k> 1. Suppose P is elliptic, i.e., the senior term P2k of P satisfies P2k(X) ~ Cp X 2k, for some constant Cpo Let ifJ be real analytic in {Ix\ < R}, and tJ.k(PifJ) = 0 at least In

a neighborhood of the origin). Then, R :$ C(P,n) < +00, where C 1.8 a constant depending on the polynomial P and the dimension of the ambient space.

Remark. The assumption in the theorem that P is elliptic is equivalent to the condition that the set {P = O} is bounded in JRn •

Corollary 7.2. Assume an is contained in the set {P = A}, a bounded algebraic set in ~n. Then, if a solution of the DP (1.1) with data x 2 is entire, n must be an ellipsoid.

PROOF. Suppose not, so deg(P) = 2k > 2, and the following Fischer decomposition) holds: \x1 2 = PifJ + 'U, tJ.'U = O. Hence, tJ.k(PifJ) = 0 and q, cannot be analytically continued beyond a finite ball of radius R = C P) < 00, a contradic~a 0

Caution. We want to stress again that, unfortunately, the theorem still tells us nothing for say small perturbations of the circle by a nonelliptic term of degree ~ 3, e.g., x2 + y2 -1 + e:(x3 - 3xy2).

8. Back to ]R2: lightning bolts

Return to the ~2 setting and consider as before our boundary an of a domain n as (part of) an intersection of an analytic Riemann surface r in (:2 with JR2 • Roughly speaking if sayan is a subset of the algebraic curve r := { x, y) : q, x, y = O}, where ifJ is an irreducible polynomial, then r = {(X, Y) E C2 : q,(X, Y = a}. Now look at the Dirichlet problem again in the context of the Goursat problem: Given, say, a polynomial data P, find f, 9 holomorphic functions of one variable near r a piece of r containing an s;; r n ~2) such that

(8.1) 'U = f(z) + g(w)\r = P(z, w),

where we have made the linear change of variables z = X +iY, w = X -iY (so iii = z on JR2 = {(X, Y) : X, Y are both real}). Obviously, .6.u = 402 azaw = 0 and u matches P on an. Thus, the DP in ]R2 has become an interpolation problem in C2 of matching a polynomial on an algebraic variety by a sum of holomorphic functions in each variable separately. Suppose that for all polynomials P the solutions u of (8.1) extend as analytic functions to a ball En = {\zI2 + \w 2 < Rn} in C2. Then, if r n Bn is path connected, we can interpolate every polynomial P(z, w) on r n Bn by a holomorphic function of the form f(z) + g(w). Now suppose we can produce a compactly supported measure p. on r n En which annihilates all functions of the form f(z) + g(w), f,g holomorphic in En and at the same time does not annihilate all polynomials P(z, w). This would force the solution u of (8.1) to have a singularity in the ball Bn in C2

• Then, invoking a theorem of Hayman [17] (see also [20D, we would be able to assert that 'U cannot be extended as a realanalytic function to the real disk B R in JR2 containing n and of radius ~ ..fiR. An

SEARCH FOR SINGULARITIES OF SOLUTIONS TO THE DIRICHLE1' PROBLEM 129

example of such annihilating measure supported by the vertices of a "quadrilateral" was independently observed by E. Study [38), H. Lewy (27), and L. Hansen and H. S. Shapiro (16). Indeed, assign alternating values ±1 for the measure supported at the four points Po:= (Zl,WI), qo: (Zl,W2), PI:= (Z2,W2), and ql:- (Z2,Wl). Then f(f +g) dl-' = l(zl) +g(wr} - I(zr) -g(W2)+ 1(Z2)+g(w2) l(z2) -g(WI) = 0 for all holomorphic functions 1 and 9 of one variable. This is an example of a closed lightning bolt (LB) with four vertices. Clearly, the idea can be extended to any even number of vertices.

Definition. A complex closed lightning bolt (LB) of length 2( n + 1) is a finite set of points (vertices) Po, qo, PI, ql, ... ,Pn, qn, Pn+l, qn+l such that Po = Pn+l, and

complex line connecting p, to q, or qj to PHI has either Z or W coordinate fixed and they alternate, i.e., if we arrived at Pj with W coordinate fixed then we follow to q, with z fixed etc.

For . domains lightning bolts were introduced by Arnold and Kolmogorov in the 19508 to study Hilbert's 13th problem (see [24] and the references therein).

The following theorem has been proved in [4] (see also [5]).

Theorem 8.1. Let n be a bounded simply connected domain in C ~ ]R2 such tlwt the Rlemann map 41: n -+ ill> = {Izl < I} is algebraic. Then all solutions of the DP wdh polynomrol data have only algebraic singularities only at branch points of t/J Wtth the branching order of the fonner dividing the branching order of the latter JJ t/J-l is a rational function. This in turn is known to be equivalent to n being a quadrature domain.

IDEA OF PROOF. The hypotheses imply that the solution u = f + 9 extends ~

as a single-valued meromorphic function into a C2-neighborhood of r. By another theorem of [4], one can find (unless 41-1 is rational) a continual family of closed

~

LBs on r of bounded length avoiding the poles of u. Hence, the measure with alternating values ± 1 on the vertices of any of these LBs annihilates all solutions u =

~

J z + g(w) holomorphic on r, but does not, of course, annihilate all polynomials of z,w. Therefore, 41-1 must be rational, i.e., 0 is a quadrature domain [36]. 0

The second author [28] has recently constructed some other examples of LBs on complexified boundaries of planar domains which do not satisfy the hypothesis of Render's theorem. The LBs validate Conjecture 3.3 and produce an estimate regarding how far into the complement C \ n the singularities may develop. For instance, the complexification of the cubic, 8x(x2 - y2) + 57x2 + 77y2 - 49 = 0 has a lightning bolt with six vertices in the (nonphysical) plane where z and ware real, i.e., x is real and y is imaginary (see Figure 3 for a plot of the cubic in the plane where x and y are real and see Figure 4 for the "nonphysical" slice including the lightning bolt). H the solution with appropriate cubic data is analytically continued in the direction of the closest unbounded component of the curve defining ao, it will have to develop a singularity before it can be forced to match the data on that component.

9. Concluding remarks, further questions

In two dimensions one of the main results in [4] yields that disks are the only domains for which all solutions of the DP with rational (in x, y) data v are rational. The fact that in a disk every DP with rational data has a rational solution was

L30 D. KHAVINSON AND E. LUNDBERG

, 2

-6 -4 -2

" -2

-3

FIGURE 3. A Maple plot of the cubic 8x(x2 - y2) + 57x2 + 77y2 -49 = 0, showing the bounded component and one unbounded component (there are two other unbounded components further away.

FIGURE 4. A lightning bolt with six vertices on the cubic 2(z + w)(z2 + w2) + 67zw - 5(z2 + w2) = 49 in the nonphysical plane with z and w real, i.e. x real and y imaginary.

observed in a senior thesis of T. Fergusson at U. of Richmond [31]. On the other hand, algebraic data may lead to a transcendental solution even in disks (see [10], also cf. [12]). In dimensions 3 and higher, rational data on the sphere (e.g., tJ == l/(Xl - a), lal > 1) yields transcendental solutions of (1.1), although we have not been able to estimate the location of singularities precisely (cf. [10]).

It is still not clear on an intuitive level why ellipsoids play such a distinguished role in providing "excellent" solutions to DP with "excellent" data. A very similar question, important for applications, (which actually inspired the program launched in [22] on singularities of the solutions to the DP) goes back to Raleigh and concerns


singularities of solutions of the Helmholtz equa.tion ([,6, - >.21u = 0, >. e JR) instead. (The minus sign will guarantee tha.t the maximum principle holds and, consequently, ensures uniqueness of solutions of the DP.) To the best of our knowledge, this topic remains virtually unexplored.

References

1. D. H. Armitage, The LAnchlet problem when the boundarv function is ennre, J. Math. Anal. Appl. 291 (2004), no. 2, 565 577.

2. S. Axler, P. Gorlon, and K. Voss, The Dlnchlet problem on quadratic surfaces, Math. Compo TS (2004), no. 246, 637 651.

3. J. A. Baker, The DIrIChlet problem for ellipsoids, Amer. Math. Monthly 106 (1999), no. 9, 829 834.

.. S. R. Bell, P. Eben£elt, D. Khavinson, and H. S. Shapiro, On the classical Dlnchlet problem In the plane unth ranonal data, J. Anal. Math. 100 (2006), 157-190.

s. , Algebralcdllm the Dlnchlet problem in the plane with rational data, Complex Var. Elliptic Equ. 52 (2007), no. 2-3, 235 - 244.

6 M Chamberland and D. Siegel, Polynomial solutions to Dirichlet problems, Proc. Amer. Soc. 129 (2001), no. 1,211-217.

7 P J Davis, The Schwarz/uncnon and Its applications, Vol. 17, Math. Assoc. America, Buffalo, NY,1974 Carns Math. Monogr.

8. P L. Duren, Polynomials orthogonal over a cUr'IJe, Michigan Math. J. 12 (1965),313-316. 9 P. Eben£elt, Smgulantles encountered by the analytic continuation of solutions to Dirichlet's

p'"Db eill, Complex Variables Theory Appl. 20 (1992), no. 1-4,75-91. 1 P D. Khavinson, and H. S. Shapiro, Algebraic aspects of the Dirichlet problem,

Quadrature Domains and their Applications, Oper. Theory Adv. Appl., vol. 156, Birkhauser, BMeI, 2005, pp 151-172.

11. P. Ebenfelt and H. S. Shapiro, The Oauchy-Kowalevskaya theorem and generalizations, Comm Partial Differential Equations 20 (1995), no. 5-6, 939-960.

12. P Ebenfelt and M. VJ.SCaJ'di, On the solution of the Dirichlet problem with rational holomorpme boundary data, Comput. Methods Funct. Theory 5 (2005), no. 2,445-457.

13. E. Fischer, tiber die der Algebra, J. Reine Angew. Math. 148 (1917), 1-78.

14. A. Friooman, Partial d,fferential equations, Holt, Rinehart and Winston, New York, 1969. 15 L Garding, T. Kotake, and J. Leray, Unif01111isation et deve!oppement asymptotique de la

B lunon du prob!eme de Oauchy lineaire, a donnees ho!omorphesj analogie avec la tMorie des ondes asymptotaques et approcMes (Prob!eme de Oauchy, Ibis et VI), Bull. Soc. Math. Frant:e 92 (1964), 263-361.

16 L. J. Hansen and H. S. Shapiro, Functional equations and hc:nmonic e:ctensions, Complex Vanables Theory Appl. 24 (1994), no. 1-2, 121-129.

17 W K Hayman, Power senes expansions for hal1nonic functions, Bull. London Math. Soc. 2 1970), 152 158.

18. G. Johnsson, The Oauchy problem m eN for linear second order partial differential equations wtth. data on II quadrIC surface, Trans. Amer. Math. Soc. 844 (1994), no. I, 1-48.

19 D. KhavIDson, Smgulantiea of hcmnomc funcnons in en, Several Complex Variables and Complex Geometry, Part 3 (Santa Cmz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Soc., Providence, Rl, 1991, pp. 207-217.

20 HoIoIIW1'Phu; parttal dlfferentlal equatsons and classical potential theorv, Universidad de La Laguna, Departamento de Ana,1isjs Matematico, La Laguna, 1996.

21. D. Khavinson and H. S. Shapiro, The Schwarz potennal in an and Oauchy's problem for the Laplo.ce equatwn, Technical Report TRlTA-MAT-1989-36, Royal Institute of Technology, Stockholm.

22. , D,,~chlet' B p,..,blem when the data '" an entire /unction, Bull. London Math. Soc. 24 1992), DO. 5, 456 468.

23. D Khavinson and N. Stylianopoul08, Rec,."ence relatIOns for orthogonal polynomials and the Kha'lllnBOII ShaPIro conJecture, in preparation.

24. S Ya. Khavinson, Best appro:nmat,on by Imear auperponnons (approzimate nomography), Transl. Math. Monogr., vol. 159, Amer. Math. Soc., Providence, RI, 1997.


26. L. Lempert, Recursion for orthogonal polynomIals on comple3: domams, Fourier Analysis and Approximation Theory, Vol. 11 (Budapest. 1976), Colloq. Math. Soc. Janos Bolyai. voL 19. North-Holland, Amsterdam, 1978, pp. 481 494.

26. J. Leray, Uniformisation de Ia solution du probleme IIn~alre analyttqUe de Cauchy pres de 10 1JarieU qui porte les donn~es de Cauchy, C. R. Acad. Sci. Paris 245 (1957). 1483-1488.

27. H. Lewy, On the reflectton laws of second order dliferenttai equations In two mdependent variables, Bull. Amer. Math. Soc. 65 (1959),37 -58.

28. E. Lundberg, Dirichlet's problem and compl~ llghtmng bolts, Comput. Methods Fund Th&ory 9 (2009), no. 1, 111-125.

29. M. Putinar and N. S. Stylianopoul08, Finite-term relatIons for planar orthogonal polynmmals. Complex Anal. Oper. Theory 1 (2007), no. 3,447 456.

30. H. Render, Real Bargmann spaces, Fischer decomposItIons, and sets of untqueneBB for poly-harmonic junctions, Duke Math. J. 142 (2008), no. 2, 313-352.

31. W. T. Ross, private communication. 32. M. Sakai, Null quadrature domains, J. Analyse Math. 40 (1981). 144-154 1982. 33. B. Yu. Sternin and V. E. Shatalov, Legendre umformlZatlon of mulhllalued analytu! junctwns.,

Mat. Sb. (N.S.) 113(155) (1980), no. 2(10), 263-284 (Russian). 34. B. Yu. Sternin, H. S. Shapiro, and V. E. Shatalov. Schwarz potentIal and smgulanUes of the

solutions of the branching Cauchy problem, Dokl. Akad. Nauk 341 1995. no. 3, 307-309 (Russian).

35. P. K. Suetin, Polynomials orthogonal over a region and Bieberbach polynom ols, Proc. Steklov Inst. Math., vol. 100, Amer. Math. Soc .• Providence, R.I., 1974.

36. H. S. Shapiro, An algebraic theorem of E. Fischer, and the holomO'FJlh Goursol problem, Bull. London Math. Soc. 21 (1989), no. 6, 513-537.

37. ___ , The Schwarz junction and its generalIZation to hIgher dlmenstons, Univ. Arkansas Lecture Notes Math. Sci., vol. 9, Wiley, New York, 1992.

38. E. Study, Einige Elementare Bemerkungen uber den Prozess der analyhschen Fortzetsung, Math. Ann. 63 (1907),239-245.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTH FLORIDA, 4202 E. FOWLER AVE.,

TAMPA, FL 33617, USA

E-mail address, D. Khavinson: dkhavins«lcas. usf . edu E-mail address, E. Lundberg: elundberlDmail. usf . edu

C~ntre d. Recherche. MatMmatlquee CRM Proceedings &nd Lecture Note. Volume 111, :iOIO

Invariant Subspaces of the Dirichlet Space

Omar El-Fallah, Karim Kellay I and Thomas Ransford

ABSTRACT. We present an overview of the problem of describing the invariant subspa.ces of the Dirichlet space. We also discuss some recent progress in the problem of characterizing the cyclic functions.

1. Introduction

Let X be a Banach space of functions holomorphic in the open unit disk [J),

such that the shift operator S: I(z) t--+ zl(z) is a continuous map of X into itself. An mvanant subspace of X is a closed subspace M of X such that 8M C M. Given / E X, we denote by [J]x the smallest invariant subspace of X conta.ining /. namely

[J]x = {pI: p a polynomial}.

We say that / is cyclic for X if [fl X = X.

1.1. The Hardy space. This is the case X = H2, where

H2:= I(z) = L akzk I IIIII~~ := L)akl2 < 00 .

k~O k~O

Recall that, for every function IE H2 \ (0), the radial limit r«() := lilD.r-+l- I(r() exists a.e. on the unit circle 'lI.'. The function I has a unique factorization I = ()h, where (),h are H2-functions, () is inner (this means that I()*I = 1 a.e.), and h is outer which mea.ns that loglh(O) I = (27l")-1 f~7I" loglh* «()lld(l). The inner factor can be expressed as product of a Blaschke product and singular inner factor. More precisely, we have () = cBSa , where c is a unimodular constant,

an E [J), L)l-lan l) < 00, lanl/an := 1 if an = 0 , n

2000 MathematZC8 SubJect Classification. Primary 47A15; Secondary 30C85, 46E20, 47B37. Ketl wordB and phrases. Dirichlet space, invariant subspace, cyclic function, logarithmic

capacity. The research of the third author was partially supported by grants from NSERC (Canada),

FQRNT (Quebec) and the eanMa. rese<ITch chairs program. This is the final form ofthe pa.per.

@2010 American Mathematlc .. l Society

133

134 O. EL-FALLAH ET AL.

and

(0' ~ 0,0' .l d9).

For more details, see for example [8,11, 13}. The invariant subspaces of H2 are completely described by Beurling's theorem

[2]:

Theorem 1. Let.M "1= (0) be an invariant subspace 01 H2. Then.M = 9J{l, where 9 is an inner function.

This result leads immediately to a characterization of cyclic functions for J{l.

Corollary 2. A function 1 is cyclic for H2 il and only il it is an O'Uter /unctwn..

1.2. The Dirichlet space. The Dirichlet space is defined by

'0:= {f(Z) = Lakzk: 1I111~:= L(k + 1) ak 2 < oo}. k~O k~O

Clearly V is a Hilbert space and V C H2. It is called the Dirichlet space because of the close connection with Dirichlet integral:

'O(f):= ~ 1 1f'(z)12 dA(z) = Lk ak 2.

D k~O

Thus 1I111~ = 111111-. + 'O(f). (The H2-norm. is added to ensure that we get a genuine norm..)

Here are two other formulas for the Dirichlet integral. The first, due to Douglas [7}, expresses the integral purely in terms of r, and leads to the notion of Besov spaces:

'O(f) = (~)2 r 1 \1*«(1) - 1*«(2) 2 d(1 d~. 271' iT T 1(1 - (2\2

The second formula is due to Carleson [5). Using the factorization above (f = cBS"h), Carleson's formula expresses V(f) in terms of the data h·, (an) and 0'. More precisely:

(1) V(f) = ~ 1 " 1 -lan \2Ih$«()\2Id(\ 271' T L.J I( - an l2

n

+ 2~ L L $1 ~ (2\2 \h$«(I)\2 \d(1\ dO'«(2)

+ _1 r r (lh$«(1)12 - \h$ «(2)\2)(log\h$ «(1)\ -loglh$«(2)$ \d~1 \d~l. 471'2 iT iT 1(1 - (2\2

It follows directly from Carleson's formula that, if 1 E V, then its outer factor h also belongs to V and satisfies 'O(h) ~ 'O(f). A further consequence is that the only inner functions which belong to V are finite Blaschke products.

In [15, 16}, Richter established an analogue of Beurling's theorem for the Dirichlet space. To state his result, we need to introduce a family of Dirichlet-type spaces V(p.). Given a finite positive Borel measure p. on the unit circle, we define

'O(p.) := {I E H2 : '01J(f) := ~ iull'(z)12cplJ(z) dA(z) < oo}.

INVARIANT SUBSPACES OF THE DIRICHLET SPACE 135

where <p,. is the Poisson transform of J.L, namely

1 2'11' 1 - IzI2 <p,. (z) := 271' 0 I( _ zl2 dJ.L( ().

We equip V(J.L) with the norm 1\·11,. defined by Hfll! := IIfll~2 + V,.(J). Note that the classical Dirichlet space corresponds to taking J.L to be normalized Lebesgue measure m on T. The following theorem was proved by Richter [15,16].

Theorem 3. Let M f. (0) be an invariant subspace of 'D. Then there exists 1 E 'D such that M e SM = Cl and M = [f]l) = fV(lfl 2 dm).

In particular, the invariant subspaces of V are all cyclic (of the form [f]l) for some 1 E V). The next theorem, due to Richter and Sundberg [17], goes one step further, expressing such subspaces in terms of invariant subspaces generated by an outer function.

Theorem 4. Let f E V \ {O}, say 1 = (}h, where () is inner and h is outer. Then

[I]l) = 9[h]l) n V = [h]l) n 9H2.

This still leaves us with the problem of describing [h]l) when h is outer. In particular, it leaves open the problem of characterizing the cyclic functions for V.

2. Dirichlet space and logarithmic capacity

Given a probability measure J.L on 'll', we define its energy by

1 I(J.L) := log Iz _ wi dJ.L(z) dJ.L(w).

Note that I(J.L) E (-00,00]. A simple calculation (see [3, p. 294]) shows that

n>l -Hence in fact I(J.L) ~ 0, with equality if and only if J.L is normalized Lebesgue measure on T.

Given a Borel subset E of 'll', we define its capacity by

c(E)

:= 1/ inf{I(J.L) : J.L is a probability measure supported on a compact subset of E}.

Note that c(E) = 0 if and only if E supports no probability measure of finite energy. It is easy to see that

countable ;- capacity zero ;. Hausdorff dimension zero

;. Lebesgue measure zero.

None of these implications is reversible. A property is said to hold quasi-everywhere (q.e.) on T if it holds everywhere outside a Borel set of capacity zero.

The following result, due to Beurling [1], reveals an important connection between capacity and the Dirichlet space.

Theorem 5. Let f E V. Then r«() := limr-+l f(r() exists q.e. on '11', and

(2) (t ~ 411/11l».


The inequality (2) is called a weak-type inequality for capacity. The strong-type inequality for capacity is

(3) 100

c(lrl ~ t) dt2 $ elll ~, where C is a constant. For a proof, see for example [191. In Theorem 15 below, we shall exhibit a 'converse' to the strong-type inequality.

Given a Borel subset E of 11', we define

V E := {I E V: r = 0 q.e. on E}.

The following result is essentially due to Carleson [41. The simple proof given be ow is taken from [3], where it is attributed to Joel Shapiro.

Theorem 6. V E is an invariant subspace 01'0.

PROOF. We just need to show that V E is closed in V, the rest is lear. Let (In) be a sequence in V E and suppose that In --+ I in V. By 2, if t > 0, then, f r all n sufficiently large,

16 I J 2 c(En{lrl~t})$c(lr-I:I~t)$ ~ n'D.

Letting n -+ 00 and then t -+ 0, we deduce that r = 0 q.e. on E. 0

Corollary 7. II I E V, then [fl'D C VZ(f") , where Z r .- ~ E T : r«) = O}.

3. Cyclic functions for the Dirichlet space

Recall that a function I E V is cyclic for V if [!l'D = V. So, fr m Corollary 7, if I is cyclic for V, then I is outer and c(Z(r)) = 0. In [31, Brov .. n and Shields conjectured that the converse is also true. In this section we will give some sufficient conditions to ensure cyclicity in the Dirichlet space. For simpliClty, we restrict our attention mostly to functions in the disk algebra A(ID). To help state these results, we introduce a class C.

Definition 8. The class C consists of closed subsets E of the unit circle satisfying the following property: every outer function I E V n A ID) such that Z feE is cyclic for V.

For functions in V n A(JI))), the Brown Shields conjecture thus becomes:

Conjecture 9. II E is a closed subset olT with c(E) = 0, then E E C.

It was proved by Hedenmalm and Shields [12] that every countable closed subset of 11' belongs to C. Richter and Sundberg [18] subsequently obtained a more general version of this result which also covered the case of functions not necessarily in A(JI))). The first examples of uncountable sets in C were given by the present authors in [9,10], and our aim is now to give an overview of this work.

Let reT be a union of disjoint arcs. We denote by ar the boundary of r in the unit circle. Given an outer function I, we associate to it the outer function II' defined by

II'(z) := exP(2~ l ~ ~: lOglr«)lld(I).

The following lemma can be obtained from Carleson's formula (1) (for the details, see [10]).


Leuuna 10. Let J E V be a bounded outer function. There is a constant C, which depends only on J, such that, lor every union 01 disjoint arcs r 01 T, and every /unction 9 E V satisfying Ig"«()1 s d«(,8r) a.e., we have IrY E V and

Theorem 11. Let I E V be an outer /unction, and set E := {( E T : liminf"'-+cl/(z)1 = o}. II 9 E 'D and Ig"«()1 S d«(, E) a.e. on T, then Y E [llD'

IDEA OF PROOF. We employ a. method developed by Korenblum [14]. Let ('Ykk:~l be the connected components ofT\E, and set r n := Uk~n'Yk' First we prove thatglr" E [fh, for all n. Then, using Lemma 10, we show that sUPnllg/r"II'D < 00.

Therefore we can extract a. subsequence of (g/r,,) that converges weakly to g. It follows that 9 E [f]1). 0

Note that the existence of a. function 9 E H2 \ (0) such that Ig*«()1 $ d«(,E) implies that

4 log(ljd«(,E)) Id(1 < 00. T

A closed subset E of l' satisfying (4) is known as a Carles on set. Let E c l' be a Carleson set. We denote by IE the outer function associated

to the dist.ance function d( (, E):

5 1 fE(z) := exp ~'

271'

Using Carleson's formula (1), it is not difficult to prove that IE E 'D. As consequences of Theorem 11, we obtain the following results.

Coronary 12. Let E be a Carleson set. Then E E C il and only il IE is cyclic forV.

Corollary 13. lIE, FE C and il E is a Carleson set, then E U FE C.

IDEA OF PROOF. Let I be an outer function in 'D n A(IOl) such that Z(J) C E u F. We have to prove that I is cyclic for 'D. Let r n = Uk~ .. 'Yk' where the ('Yk) are the connected components of l' \ E. We can write

IIi; = Ir"fEhv"IE'

Note that 8(1' r .. ) is a finite set. Since F E C, we have F U 8(11' \ r .. ) E C. From the fact that Z(iT r"fE) C F U 8(1' \ rn), we deduce that h\l""IE is cyclic in 1). Thus Ir"IE E [1]1). Using Lemma 10, we obtain that Il/r"IEI\'D is uniformly bounded, SO IE E [1]'D' Since E E C, it follows that IE is cyclic, and therefore so ~~ 0

We now turn our attention to Conjecture 9. We shall establish two partial results in this direction both of them sufficient conditioIlB for a closed set E to , ,

elong to the class C.


3.1. A capacitary sufficient condition. Given a closed set E c T and t > 0, we write E t := {( E 11' : d«(,E) ::; t}, and denote by IEt \ the Lebesgue measure of E t . It is clear that c(E) = limHO+ c(Et ). In the following theorem we prove that, if c(Et ) goes to zero sufficiently rapidly as t -+ 0+, then E E C.

Theorem 14. Let E be a closed subset of 11'. If

(6) r (E )loglog(l/t) d Jo c t t 10g(1/t) t < 00,

then E E C.

The proof of Theorem 14 is based on the following converse (proved in [9}) to the strong-type inequality for capacity (3).

Theorem 15. Let E be a proper closed subset ofT, and let 11: (0,71"1-+ R+ be a continuous, decreasing function. Then the following are equivalent:

(i) there exists h E V such that

Ih*«()1 ~ 11(d«(,E)) q.e. on 11';

(ii) there exists h E V such that

Reh*«() ~ 11 (d«(, E)) and IImh*«()I::; 71"/4 q.e. on 11';

(iii) E and 11 satisfy

(7)

IDEA OF THE PROOF OF THEOREM 14. Suppose that condition 6 holds. By Theorem 15, there exists hE V such that

Reh*«() ~ log log (d«(, E)) and IImh*(OI::; 71" 4 q.e. on T.

We consider the analytic semigroup cP). := exp( -Aeh ) for arg(A) < 71" 4. It has the following properties:

(a) the map AI-t cP). is holomorphic from {A: larg(A)\ < 71" 4} to Vj (b) limHO+I\'fJt -11\'0 = 0; (c) Icp).(z) I = O(d(z, E)) for large A.

Let f E VnA(][Jl) be an outer function such that Z(f) c E, and let 1/J be an element of the dual space of V which is orthogonal to [fl'D' It follows from property (c) and Theorem 11 that CPt E (fl'D for large t. Thus (PCPt, t/J) = 0 for every polynomial p. Using property (a), we can extend this equality to all t > O. Property (b) then implies that (p, t/J) = O. Hence t/J = 0 and f is cyclic. 0

3.2. A geometric sufficient condition. By geometric condition, we mean a condition expressed in terms of IEtl. It is well known (see for example [6}) that there exists a constant C> 0 such that, for every closed set E c 11', we have

r ds C (8) Jo lEal::; c(E)'

In particular, if f; dt/IEtl = 00, then c(E) = O. In the case of Cantor-type sets the converse is a.lso true [6}. Using (8) with E t in place of E, we obtain

1" ds C t lEal::; c(Et )'


So, from Theorem 14, if E satisfies

IEtl o (t l~g(l/t) )2 dt < 00,

then E E C. The following theorem gives a more precise condition.

Theorem 16. Let E be a closed subset of 'If such that lEt! = OW) for some f> o. If

dt

o lEt! then E E C.

'Ib give an idea of the proof of this theorem, we need to digress slightly and introduoe a generalization of the functions fE defined in (5). Let E be a closed subset of T of Lebesgue measure zero, and let w: (0, 11'] ~ jR+ be a continuous function such that

9 Ilogw(d«(,E))lld(1 < 00. T

We shall denote by f w the outer function satisfying

10 If':«()1 = w(d«(,E)) a.e.

Functions of this kind were already studied, for example, by Carleson in [4], in the course of his construction of outer functions in Ak(ll}) with prescribed zero sets. The following result gives a two-sided estimate for the Dirichlet integral of certain of functions (for the details of the proof, see [10]).

Theorem 11. Let E be a closed subset ofT of measure zero, let w: (0, 1I']-t jR+

be an lnCfw.sing function such that (9) holds, and let fw be the outer function given b 1 . Suppose further that there exists 'Y > 2 such that t t-+ w(fY) is concave. Then

11 V(fw) x w' (d(C, E))2d(C, E) IdCI, T

where the implied constants depend only on 'Y. In particular, fw E V if and only if the ntegml In (11) zs finite.

loEA OF THE PROOF OF THEOREM 16. We first suppose that E is regular, in the sense that Ed x "p(t), where "p is a function such that 1/J(t)/tOt is increasing for some Q E (!. 1).

For 6 E (0,1), define W6: (0, 11"] ~ jR+ by

6'" t1-Ot iiit6) I

A6 -logft dsN(s),

1,

0< t ~ 0,

0<t~1'/6,

1'/6 < t ~ 11'.

Here, the constants A6,1'/6 are chosen to make W6 continuous. Note that W6 is incrpaging, and one can show that t t-+ W6(fY) is concave if'Y > 1/(1 - a) and 0 is sufficiently small. Using Theorem 17, we obtain that limsuP6-+0 V(fw6) < 00.

From the condition fo dt/IEtl = 00, it is easy to see that lim6-+o 1'/6 = 0, which implies that li1n6 ,01/':',1 = 1 a.e. and that lim6-+01/w, (0)1 = 1. Putting these facts together, we deduce that fWI ~ 1 weakly in V as 0 -t O.


Now, for each 6 E (0,1), the quotient Wli(t)/t1- 0 is bounded, 80 1108//1-0 is bounded on 11)). By [17, Lemma 2.41, it follows that IW 4 E [/};-O}p. Letting 6 -+ 0, we obtain 1 E [f};-(11). Also, by [17, Theorem 4.3], we have [/1-0 ]1) = [/E]1). Hence f E is cyclic and E E C.

When E is not regular, the result is still true, but now there is an extra step in the proof. To obtain the function t/J, we first need to regularize E t , using a quantitative form of M. Riesz's rising-sun lemma_ For the details, we refer to ~~. 0

References

1. A. Beurling, Ensembles e:r;ceptionneis, Acta Math. 72 (1940), 1-13. 2. ___ , On two problems concerning lmear transformahons m Hubert space, Acta Math. 81

(1949), 239-255. 3. L. Brown and A. L. Shields, Cyclic vectors in the DIrichlet space, Trans. Amer. Math. Soc.

285 (1984), no. 1, 269-303. 4. L. Carleson, Sets of uniqueness for junctions regular In the Un1t CIrcle, Acta Math. 87 1952,

325-345. 5. ___ , A representation formula for the Dirichlet Integral, Math. Z. 73 1960,190-196. 6. ___ , Selected problems on e:r;ceptlonal sets, Van Nostrand Mathematical Studies, 'VOl 13

Van Nostrand, Princeton, NJ-Toronto, ON-London, 1967 7. J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 1931, no. 1,

263-321. 8. P. L. Duren, Theory of HP spaces, Pure Appl. Math., vol. 38, Academic Press, New York,

1970. 9. O. El-Fallah, K. Kellay, and T. Ransford, CycliCIty In the DIrichlet space, Ark. Mat. 44

(2006), no. 1, 61-86. 10. ___ , On the Brown-Shields conjecture for cyci1clty In th DInchlet space, Adv. Math.

222 (2009), no. 6, 2196-2214. 11. J. B. Garnett, Bounded analytic junctions, 1st ed., Grad. Texts m Math, vol. 236, Springer,

New York, 2007. 12. H. Hedenma.lm and A. Shields, Invariant subspaces In Banach spaces of analytic junctions,

Michigan Math. J. 37 (1990), no. 1,91-104. 13. P. Koosis, Introduction to Hp spaces, London Math. Soc. Lecture Note Ser., 'VOl. 40, Cambridge

Univ. Press, Cambridge, 1980. 14. B. I. Korenblum, Invariant subspaces of the shift operotor In a weighted Hubert space, Mat.

Sb. (N.S.) 89(131) (1972), 110-137 (Russian); English uansi., Math. USSR-Sb. 18 197'2, 111-138.

15. S. Richter, Invariant subspaces of the DIrichlet shift, J. Reine Angew. Math. 386 (1988 , 205-220.

16. ___ , A representation theorem for cychc analytic two-lSometnes, Trans. Amer. Math. Soc. 328 (1991), no. 1, 325 349.

17. S. Richter and C. Sundberg, Mulhphers and invariant subspaces In the DIrichlet space, J. Operator Theory 28 (1992), no. 1, 167 186.

18. ___ , Invariant subspaces of the DIrichlet shift and pseudocontlnuanons, Trans. Amer. Math. Soc. 341 (1994), no. 2,863 879.

19. Z. Wu, Carleson measures and muihphers for Dirichlet spaces, J. Funct. Anal. 169 (1999), no. 1, 148 163.

DEPARTEMENT DE MATHEMATIQUES, UNIVERSITE MOHAMED V, B.P. 1014, RABAT, MOROCCO E-mail addre8s:elfallahOfar.ac.ma

LABORATOIRE D' ANALYSE, TOPOLOGIE, PROBABILITES, CENTRE DE MATHEMATIQUES ET IN

FORMATIQUE, UNIVERSITE DE PROVENCE, 39, RUE F. JOLIOT-CURlE, 13453 MARSEILLE CEDE)( 13, FRANCE

E-ma'll addre8s: kellayClcm1.un1v-mrB.fr

INVARIANT SUBSPACES OF THE DIRICHLET SPACE

DEPARI'EMENT DE MATHEMATIQUES ET DE STATISTIQUE, UNIVERSITB LAVAl

Gl V OA6, CANADA E-rnln! address: 1 _ • • U &.v ...... c&.

Oentre d. R<-eherch •• Math'matlque. CRM Proceeding. and I, octure Note. Volume 111, 3010

Arguments of Zero Sets in the Dirichlet Space

Javad Mashreghi, Thomas Ransford, and Mahmood Shabankhah

ABSTRACT. We characterize the unimodular sequences (el9n )n>l such that -(rneilln )n>l is a zero set for the Dirichlet space for every positive Blaschke

(rn)n> l' The principal tool is a characterization of Carleson sets in -teIms of their convergent subsequences.

1. Introduction

The Dirichlet space V is the set of functions I, holomorphic on the open 1Init disk D, for which

1 V(f):= -

11' I!'Cz)12dxdy < 00.

D

H J z = L::'-o ClnZn , then V(f) = L::'=1 nlanl2. Hence V is properly contajned in the Hardy space

00 00

H2:= fCz) = L anzn : 11/111-2 := ~lanl2 < 00 .

n=O n=O

'D is a Hilbert space with respect to the norm II·II'D defined by [I/II~ := V(f)+11/111-2' Let X be a space of holomorphic functions of II)). A sequence CZn)n~1 C II)) is

said to be a zero sequence for X if there exists functiOIi 1 EX, not identically zero, which vanishes on zn, n ~ 1. We do not require that the (zn) be the only zeros of J. H Zn n>l is not a zero sequence for X, then we call it a uniqueness sequence -for X.

•

Zero sequences of the Hardy space H2 are completely characterized: a sequence z..)n>l c D is a zero sequence for H2 if and only if it satisfies the Blaschke condition -1)

00

LC1-lzn\) < 00.

n=1

2000 Math.emanu SubJect Classificat1.on. Primary 30Cl5j Secondary 30D50, 30D55, 31C25. Kf!1I words and phrases. Zero sets, Dirichlet space, Carleson sets, Blaschke sets. Fitst author partially supported by NSERC (Canada), FQRNT (Quebec). Second author partially supported by NSERC, FQRNT and the Canada research chair!

prOgJam. Third author supported by an NSERC Canada graduate scholarship. Tills is the final fOlm of the paper.

@>2010 American M"them"tlc .. 1 Socletl

143

144 J. MASHREGHI ET AL.

Since 1) C H2, this is also a necessary condition for a zero sequence for 1).

However, it is far from being sufficient. Indeed, the complete characterization of the zero sequences of 1) is still an open problem.

The first breakthrough in this direction was the pioneering work of Carleson [4]. He showed that if a sequence (Zn)n~1 in 11) satisfies

00 ( 1 )1-11 ~ -log(l-Iznl) < 00

for some e > 0, then (Zn)n~1 is a zero sequence for V. Using a completely different approach, Shapiro and Shields [10] obtained showed that this result remains true even with e = O.

Thus, if a sequence (rn)n~1 in [0,1) satisfies

00 1 '" <00 ~ -log(l- rn) ,

(2)

then (rnei6n )n>1 is a zero sequence for V for every choice of (e'B .. )n> . Later on, Nagel, Rudin and Shapiro [8] showed that, if (2) is not satisfied then there exists a sequence (ei6n )n>1 for which (rnei6n )n>1 is a uniqueness sequence f r V. Putting

these results tog~ther, we conclude th;t (rn ei6 .. )n>1 is a zero sequence f r V f r

every choice of (ei6n )n>l if and only if (2) holds. -The main purpose -of this article is to prove the following theorem, which can

be considered as the dual of the last statement. It was already stated as a remark in [6, p. 704, line 12], but without detailed proof.

Theorem 1. Let (ei6 .. )n~1 be a sequence in'll'. Then (r e'B n~l 1S a zero se

quence for 1) for every positive Blaschke sequence (rn)n~l if and nly J {elll :n ~ I} is a Carles on set.

We recall that a closed subset F of 'll' is a Carleson set if

(3) llOg dist«, F) \d(\ > -00,

where "dist" is measured with respect to arclength distance. These sets were first discovered by Beurling [1] and then studied thoroughly by Carleson. Carleson [3} showed that the condition (3) characterizes the zero sets of f T for f E AI, where A 1 := {f E 0 1 (iOi) : f is holomorphic on Ill>}. If F is a closed subset of T of Lebesgue measure zero whose complementary arcs are denoted by In, n ~ 1, then (3) is equivalent to

(4) n

Our proof of Theorem 1 is based on the following theorem, which we believe is of interest in its own right. In particular, it characterizes Carleson sets in terms of their convergent subsequences.

Theorem 2. Let E be a subset ofT. Then E is a Carleson set if and only if the closure of every convergent sequence in E is a Carleson set.

Theorem 2 is proved (in Ito more general form) in Section 2, and Theorem 1 is deduced from it in §3. Finally, in §4 we relate these results to the notion of Blaschke sets.

ARGUMENTS OF ZERO SETS IN THE DIRICHLET SPACE 145

2. Proof of Theorem 2

We shall in fact prove Theorem 2 in a more general form. Let w : [0,1T] -+ [O,oo} be a continuous, decreasing function such that w(O) = 00 and Io'" w(t) dt < 00. A closed subset F of l' is 8.J) w-Carleson set if

(5) w( dist((, F)) Id(\ < 00. T

IT F is a closed subset of l' of measure zero, then condition (5) is equivalent to

00 11nl/2 L w(t) dt < 00,

n=1 0

where (In)n~l are the components of1'\F (see [7, Proposition A.I]). The classical Carleson sets correspond to the case w(t) == log+(l/t), and so Theorem 2 is a special case of the following result.

3. Let w be as above and let E be a subset of 1l'. Then E is an w-Carleson set if and only if the closure 0/ every convergent sequence in E is an w-Carleson set.

PROOF. Set F = E. We need to show that F is an w-Carleson set. We first show that the closure of every convergent sequence in F is an w

Carleson set. As the lInion of two w-Carleson sets is again one, it suffices to consider sequences converging to a limit from one side. Suppose that (e iBn ),,>1 C F, where

-81 < 82 < ... and 8n -+ 90 as n -+ 00. Since lim.,-to+ I; w(t) dt = 0, we may choose a positive sequence (7],,),,>11 such that -

00 '7n

L w(t) dt < 00.

,,=1 0

Set 5 := mjn{7]n, 7]n-1, (9"+1 - 9,,)/2, (9" - 9,,_1)/2}, n ~ 2, and choose eitPn E E ,th tP E 9n - 8", 9n + 8,,}, n ~ 2. Then ¢2 < ¢3 < . " and ¢n -+ 90 as n -+ 00.

SUlce eltP )n>2 is a convergent sequence in E, it follows from the assumption that -

e'tP : n ~ 2} is an w-Carleson set, and therefore

w(t) dt < 00.

n=2 0

AB 8 +1 - On $ tPn+1 - ¢n + 27]n and w(t) is a decreasing function, (8 +1-8,,)/2 (tPn+l-tPn)/2 '7n

w(t) dt :::; w(t) dt + w(t) dt. o o 0

Thelefore

L n>1 0

(8,,+1-8,,)/2

w(t) dt < 00,

and {e'9n : n :::: I} is an w-Carleson set. We next show that F bas zero Lebesgue measure. Suppose, on the contrary,

that F > O. We claim that there exists a positive sequence (en ),,>1 such that -6) and

00

L n 1 0

1! .. /2 w(t) dt == 00.

J. MASHREGHI ET AL. 146

Indeed, since (l/x) J; w(t) dt -+ 00 as x -+ 0+, we may choose integers N., i ~ 1,

such that iFl/(2i+1 N.)

fa w(t) dt ~ I/Ni (i ~ 1).

Then, the sequence

satisfies (6). Now choose (On),,;:::1 inductively as follows. Pick 01 such that ri-91 E F. If 01 , ••• ,On have already been selected, choose On+1 as small as possible such that On+1 ~ On + en and ei9,,+1 E F. Note that On < 01 + 211' for all n, for otherwise F would be covered by the finite set of closed arcs [ei9, ,ei(9,H, 1;=1' contradicting

the fact that 22';=1 ej < IFI. Thus (ei9" )n;:::l is a convergent sequence in F. Also

1(9n+1 -9,,)/2 1En/2 L w(t) dt ~ L w(t) dt = 00,

nOn 0

so {ei9n : n ~ I} is not an w-Carleson set. This contradicts what we proved in the previous paragraph. So we conclude that IFI = 0, as claimed.

Finally, we prove that F is an w-Carleson set. Once again, we proceed. by contradiction. If F is not an w-Carleson set, then, as it has measure zero, it follows that

(7) 00 (11nl/2

L J~ w(t) dt = 00, n=l 0

where (In)n;:::1 are the components of 1l'\F. Denote by ei9n the midpoint of 1m where On E [0,211']. A simple compactness argument shows that there exists 0 E [0,211') such that, for all 8 > 0,

111,,1/2 L w(t) dt = 00.

9nE(9-o,9+o) 0

We can therefore extract a subsequence (In,) such that Onj -+ 0 and

111",1/2

~ 0 w(t)dt = 00.

3

The endpoints of the In; then form a convergent sequence in F whose closure is not an w-Carleson set, contradicting what we proved earlier. We conclude that F is indeed an w-Carleson set. 0

3. Proof of Theorem 1

If {ei9n : n ~ I} is a Carleson set then, by [9, Theorem 1.2), for every Blaschke sequence (rn)n;:::1 in [0, 1), there exists f EAoo, f t=0, which vanishes on (rnei9n )n~l' Here Aoo := {f E Coo(D) : f is holomorphic on !Ill}. In particular, (rnei9,,) >1 is a n_ zero sequence for V.

For the converse, we use a technique inspired by an argument in [5]. Suppose that (rnel9n )n;:::1 is a zero sequence for V for every Blaschke sequence (rn),,>1 C

ARGUMENTS OF ZERO SETS IN THE DIRICHLET SPACE 141

[0,1). Let (ei9ft#) ">1 be a convergent subsequence of (eI9ft) n>I' We shall show

~a-"-----7'- -that {ei9ftl : j ~ I} is a Carleson set. As the union of two Carleson sets is again a Carleson set, it suffices to consider the case when (In! < (Jn~ < .,' and (JnJ ~ (Jo,

as ; -+ 00. Let OJ := (Jnl+l - (Jnl' j ~ 1. Consider the Blaschke sequence

(j ~ 1).

By hypothesis, this is a zero sequence for V, and as such it therefore satisfies

2". 00 l-lz"12 log L 1 2 d(J < 00,

o lel9 - z"1 3=1 3

(see for example [5, p. 313, equation (2)]). On the other hand, for 9 E (9n~' 9nk+1),

we have lel8 - Zk 1 :5 20k which gives

1 - IZkl2 1 " 2 ~ I lel8 - z,. I 40,.

and consequently

2..- 00 2 00

og 2 o 3=1 lel8 - Zjl d(J~L

k=1

We conclude that E", OJ log Ok > -00. Thus {el9nk : k ~ I} is indeed a Carleson set. Now apply Theorem 2 to obtajn the desired result. 0

4. Blaschke sets

Let X be a space of holomorphic functions on lIJl and let A be a subset of lIJl. We say that A is a Blaschke set for X if every Blaschke sequence in A (perhaps with repetitions) is a zero sequence for X. Blaschke sets for Aoc were characterized by Taylor and Williams [11], and for V by Bogdan [2]. The following theorem summarizes their results and takes them a little further.

Theorem 4. Let A be a subset of lIJl. The following statements are equivalent.

(a) A is a Blaschke set for V. (b) A is a Blaschke set for Aoo, (c) Every convergent Blaschke sequence in A is a zero sequence for 'D. (d) Every convergent Blaschke sequence in A is a zero sequence for Aoo. (e) For every Blaschke sequence (rnel9n )n>1 in A, {el9n : n ~ I} is a Car-

Leson set. -(f) The Euclidean distance dist«(, A) satisfies

10gdist«(,A) Id(1 > ..... 00. T

PROOF. The equivalence of (a), (b), (e) and (f) was already known. Indeed, a).: :. (f) is Bogdan's theorem [2, Theorem I], and (b).: :. (f) is due to Taylor

and Wi11iams [11, Theorem 1]. Also (b) < :. (e) follows from a result of Nelson [9, Theorem 1.2].

The new element is the equivalence of these conditions with (c) and (d). It is obvious that (b) :- (d) and that (d) :. (c), so it suffices to prove that (c) :. (e).

148 J. M .... SHREGHI ET AL.

Assume that (c) holds. Let (rnei6n ) be a Blaschke sequence in A, and let (e'9,,;)

be a convergent subsequence of (ei6n ). Set B := {r n; ei6n.1 : j ~ I}. Then everY

Blaschke sequence in B is a convergent Blaschke sequence in A, so by hypothesis (c) it is a. zero sequence for V. In other words B is a Blaschke set for V. By the eqUi~

lence of (a) and (e), but applied to B in place of A, it follows that {ei6nj : j ~ I} is

a Carleson set. By virtue of Theorem 2, we deduce that {ei6n : n ~ I} is Carleson set. Thus (e) holds, and the proof is complete. IJ

Remark. Theorem 2 can be deduced, in turn, from Theorem 4. Let E be a subset of '][' such that the closure of every convergent sequence in E is a Carleson set. Define

A:= {rei6 : r E [0, 1),ei6 E E}.

Let (rnei9n )n>l be a. convergent Blaschke sequence in A. Then (~9n) is a convergent

sequence in E~ so {ei9n : n ~ I} is a Carleson set. By [9, Theorem 1.2}, the sequence (rnei9n) is a zero sequence for A"". To summarize, we have shown that A satisfies condition (d) in Theorem 4. Therefore A also satisfies condition e). From this it follows easily that E is a Carleson set.

References

1. A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940),1-13. 2. K. Bogdan, On the zeros of functions with finite Dirichlet mtegml, Kodai Math. J. 19 1996,

no. 1,7-16. 3. L. Carleson, Sets of uniqueness for functions regular in the umt csrcle, Acta Math. 87 1952,

325-345. 4. __ , On the zeros of functions with bounded Dirichlet mtegrols, Math. Z. 56 1952,289-

295. 5. J. G. Caughran, Two results concerning the zeros of funcnons '\/11th fimte Dtnchlet mtegrol,

Canad. J. Math. 21 (1969), 312-316. 6. __ , Zeros of analytic functions with infinitely dtfferennable IHrundary values, Proc. Amer.

Math. Soc. 24 (1970), 700-704. 7. O. El-Fallah, K. Kellay, and T. Ransford, Cycl,csty m the Dmchlet space, Ark. Mat. 44

(2006), no. 1,61-86. 8. A. Nagel, W. Rudin, and J. H. Shapiro, Tangent,aI boundary behavtor of functions ,n

Dirichlet-type spaces, Ann. of Math. (2) 116 (1982), no. 2, 331-360. 9. J. D. Nelson, A characterization of zero seta for Aoo, Michigan Math. J. 18 (1971), 141-147. 10. H. S. Shapiro and A. L. Shields, On the zeros of functions '\/11th finde Dtnchlet mtegrol and

some related function spaces, Math. Z. 80 (1962), 217-229. 11. B. A. Taylor and D. L. Williams, Zeros of L'psch,tz funcnons analytic m the und d\8c,

Michigan Math. J. 18 (1971), 129-139.

DEPAIUEMENT DE MATHEMATIQUES ET DE STATlSTIQUE, UNIVERSITE LAVAL, 1045, AVENUE DE

LA MEDECINE, QUEBEC, QC GIV OA6, CANADA

E-mail address.J.Mashreghi:javad.mashreghiGmat.ulava.1.ca E-mail address.T.Ransford:randordlDmat.ulava.1.ca E-mail address.M.Shabankha.h:mahmood.shabankhah.llDul.ava.1.ca

Cone .... d. Reehorehea MMh6rnAtlqu,", CRM Proceeding. and I ecturo Not •• Volume Ill, 2010

Questions on Volterra Operators

J aroslav Zemanek

Consider the classical Volterra operator t

(V f)(t) = I(s) ds o

on the Lebesgue spaces £1)(0,1), 1 ::; p ::; 00, and its complex analogue

(Wf)(z) = 1(>') d>' o

on the Hardy spaces HP on the unit disc, 1 ::; P ::; 00.

The important Allan - Pedersen relation

8-1(1 - V)8 = (1 + V)-I,

where 8f)(t) = et I(t), 1 E £1'(0,1), 1 ::; p ::; 00 was noticed in [1] and extended by an elegant induction to

8-1 (1 - mV)8 = (1 - (m -1)V)(I + V)-l

m [10], for m = 1, 2, • • •. Analogously, we have the corresponding complex formulas. In fact, the formula

8-1(1 - zV)8 = (1 - (z - I)V)(1 + V)-1

is true for any complex number z. Indeed, from the Allan-Pedersen relation we have

and then

8-1(1 - zV)8 = 1 - zV(1 + V)-1 = (1 - (z - I)V)(1 + V,-I,

Since (1 + V)-llh = Ion L2(0, 1) by [5, Problem 150], it follows that every operator of the forro 1 -tV, with t ~ 0, is power-bounded on L2(0, 1). This in turn implies, as observed in [10] by using [3, Lemma 2.1] and [9, Theorem 4.5.3], that

1 1(1 - V)n - (1 - v)n+1112 = O(n-1/ 2) as n -t 00.

2000 Ma.thematU!6 Subject Clas8ification. Primary 47GlOj Secondary 47 A10, 47 A12, 47 A36 This paper was supported in part by the European Community project TODEQ (MKTD·

CT-2005-(30042). This is the final fOlm of the paper.

@)2010 American Mathematic&! Soclet)

149

150 J. ZEMANEK

The exact order of the norms of powers (I - v)n on V(O,I) and of their consecutive differences was obtained in [8], connecting the power boundedness on L2(0, 1) with the result 11(1 - v)nlh = O(n1/4) on L1(0, 1) obtained in [61.

In particular, the operator 1 - V is power-bounded on V(O, 1), 1 ::; p :5 00, if and only if p = 2. Similarly, the order (1) holds on L2(0, 1) only.

More generally, the following characterization has recently been obtained by Yu. Lyubich [7]. Let ¢(z) be an analytic function on a disc around zero such that ¢(O) = 1, ¢(z) ¥= 1. Then ¢(V) is power-bounded on V(O, 1) if and only if lP'(O) < 0 and p = 2.

If p =F 2, a sequence of functions was found in [8] on which the powers of the operator 1 - V increase correspondingly. However, by the Banach-Steinhaus theorem, plenty of single functions should exist in V(O, 1), p =F 2, on which the powers of 1 - V are not bounded.

Question 1. Find a function f in V(O, 1), p =F 2, such that

sup 11(1 - vt flip = 00. n=1,2, ...

We were not able to do that! Only indirectly, J. Sanchez-Alvarez showed that on the function

f(x) = xP- 1

with (p -1)/p < f3 < ~ and 1 :5 p < 2 we have

limsupn!II[(1 - vt - (1 - v)n+1]f P = 00, n-+oo

by considering the subsequence n = 4m2 • So (1) does not hold, hence the operator 1 - V cannot be power-bounded (the same reasoning as above that led to (1)). H p = 2, then such a f3 does not exist, thus no contradiction.

The complex operator 1 - W is not power-bounded on H2. It was observed by V.1. Vasyunin and S. Torba that the norms of the polynomials (1 - w)nl (i.e., the Euclidean norms of their coefficients) increase very fast.

Question 2. Is it possible to characterize the space H2 among the spaces BP, 1 :5 p :5 00, in terms of the growth of the operator norms of the powers of 1 - W?

The numerical range of the operator V on the Hilbert space L2 (0, 1) is described in [5, Problem 166]. Since the operator 1 + V preserves positivity of functions, the numerical ranges of all the powers are symmetric with respect to the real axis. Moreover, they are not bounded (since the operator 1 + V is not power-bounded, see [12]) and not contained in the right half-plane Rez ~ ° (since the operator V is not self-adjoint, see [4] and some references therein).

Question 3. What is the union of the numerical ranges of the powers of the operator 1 + Von L2(0, I)? Is it all the complex plane? What about the operators 1 ± W on H2? Or, even 1 - V on L2(0, I)?

The numerical ranges of (1 + v)n on L2(0, 1) were approximately determined on computer by 1. Domanov, for n = 1,2, ... ,7. It turns out that these seven sets are increasing, for n = 2 catching 1 as an interior point, and for n = 5 already reaching the negative half-plane Rez < 0.

QUESTIONS ON VOLTERRA OPERATORS 151

Question 4. Fix a half-line l starting at the spectral point 1 of the operator I + V on L2(0, 1), and denote by In the length of the intersection of l with the numerical range of (I + V)". What is the behaviour of In with respect to n? Does the limit l,,/n exist? Does it depend on the direction of l?

It is interesting to observe, as pointed out by M. Lin, that the power boundedness of operators is very unstable even on segments: the operator

(1 - 0:)(1 - V) + 0(1 + V2) = OV2 - (1 - o)V + 1

on L2(0, 1) is power-bounded for ° ::::; 0 < 1 by [11, Theorem 5], but not for 0 = 1 by [10, Theorem 3].

Question 5. Does there exist a quasi-nilpotent operator Q such that the operators 1 - tQ are power-bounded for ° ::::; t < 1, but not for t = I?

Let (Mf)(t) = tf(t)

be the multiplication operator on L2(0, 1). It was shown in [2, Example 3.3] that the Volterra operator V belongs to the radical of the Banach algebra generated by M and V. all the products in M and V, involving at least one factor V, as well as their linear combinations, are quasi-nilpotent operators (which does not to be obvious from the spectral radius formula!). Hence all these candidates can be tested in place of Q above.

Moreover, a number of variants of Question 5 can be considered for various versions of the well-known Kreiss resolvent condition (studied, e.g., in [8]). The answer is particularly elegant for the Ritt condition [10, Proposition 2].

I wish to thank the referee for his interesting comments.

References

1 G. R. AHan, Power-bounded elements and radical Banach algebras, Linear Operators (Warsaw, 1994, Banach Center Publ., vol. 38, Polish Acad. Sci., Warsaw, 1997, pp. 9-16.

2. J BraCie, R.. Dmoveek, Yu. B. Fa.rforovskaya. E. L. Rabkin, and J. Zemanek, On positive comm"tot~r8, POSItivity, to appear.

3 S R. Foguel and B. Weiss, On con'IJex power series of a conse1'1lative Marko'll operator, Proc. Amer Math. Soc. 38 (1973),325-330.

4. A. Gomdko, l. Wrobel, and J. Zemanek, Numerical ranges in a strip, Operator Theory 20, Theta Ser. Adv. Math., vol. 6, Theta, Bucharest, 2006, pp. 111-121.

5 P R. Ha1m08, A Hubert space JI'''Oblem book, Van Nostrand, Princeton, NJ-Thronto, ONLondon, 1967.

6 E. HIlle, Remarks on ergodlc theorems, Trans. Amer. Math. Soc. 57 (1945), 246 269. 7 YIL Lyubicb., The power boundedness and resol'IJent conditi.ons for functions of the classical

Vo/te" a , Studia Math. 196 (2010),41-63. 8 A. Montes-Rodriguez, J. Sanchez-Alvarez. and J. Zemanek, Un'fonll Abel-KrelBs bounded-

neff and the beha'IIWur of the VoltefJa operator, Proc. London Math. Soc. (3) 91 2(05), no. 3, 761 788.

9 O. Neva.nJinna , Con'IJergence of lteratwns for lmear equations, Math. ETH Zurich, BU'khanser, Ba<>el, 1993.

10 D Tsedenbayar, On the power boundedness of certain Voltena operator pencils, Studia Math. 156 (2003), no. 1, 59 66.

11 D. Tsedenbayar and J. Zemanek, Polynomials in the Volten'IJ and R,tt operotors, Topological Algebras, Their ApplicatioDIJ, and Related Topics, Banach Center Publ., vol. 67, Polish Acad. Set., Waxsaw. 2005, pp. 385-390.

152 J. ZEMANEK

12. J. ZemAnek, On the Gef/and-Hille theorems, Functional Analysis and Operator Theory (Warsaw, 1992), Banach Center Pub!., vol. 30, Polish Acad. Sci., Warsaw, 1994, pp. 369-385.

INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES, P.O. Box 21, 00-956 WAR!3AW 10, POLAND

E-mail address:zemanekCimpan.pl

Oontre de Recherche. MAth'm"tlquea CRM ProceedlRp And Lectu .... Notel Volume Ill, 2010

Nonhomogeneous Div-Curl Decompositions for Local Hardy Spaces on a Domain

Der-Chen Chang, Galia Dafni, and Hong Yue

ABSTRACT. Let 0 C lRn be a Lipschitz domain. We prove diy-curl type lemmas for the local spaces of functions of bounded mean oscillation on n, bmOr(O) and bmo.(O), resulting in decompositions for the corresponding local Hanly spaces h!(O) and h~(O) into nonhomogeneous div-curl quantities.

1. Div-curl lemmas for Hardy spaces and BMO on JRR

Tbis article is an outgrowth, among many others, of the results of Coifman, Lions, Meyer and Semmes ([7]) which connected the div-curl lemma, part of the theory of compensated compactness developed by Tartar and Murat, to the theory of real Hardy spaces in ]RR (see [10]). In particular, denote by HI (JRR

) the space of distributions (in fact Ll functions) f on lRn satisfying

1

f r some fixed choice of Schwartz function cf> with f cf> = 1, with the maximal function Mq, defined by

One version of the results in [7] states that for exponents p, q with 1 < p < 00, - -1 p+ l/q = 1. and vector fields V in V(lRn,JRn), W in U(JRR,JRn) with ... -divV = 0, curlW = 0

•

2000 Mathemat'ICII Subject Ciasstjication. Primary 42B30; Secondary 46E99, 30D50. KerJ wont. and phrases. Bounded mean oscillation, Hardy spaces, Lipschitz domain, div-curl

div curl decomposition. The first author is partially supported by a Hong Kong RGO competitive earmarked research

giant #600607 ami a competitive research grant at Georgetown University. The second author is partially supported by the Natural Sciences and Engineering Research

CounCIl of Canada,

The third author is partially supported by the Natural Sciences and Engineering Research Council, and the Centre de recherches matMmatiques, Montreal.

This is the final form of the paper.

@2010 Dar-Chen Chanl, Galla Oarnl, and Honl Yue

153

154 D.-C. CHANG ET AL.

in the sense of distributions, the scalar (dot) product / = V. W belongs to HI (R" . Moreover, one can bound the HI norm (defined, say, as the £1 norm of M~U ) by

IIVIIL"IIWI!LQ. While a local version of this result, in terms of Hlo." is given in [7}, in order

to obtain norm estimates we use instead the local Hardy space hI JR." • This was defined by Goldberg (see [11]) by replacing the maximal function in 1 by its "local" version

(2) m~(f)(x) = sup If * (Pt{x) . O<t<l

Again the norm can be given by IIm~(f)IIL1(1It") and different choices of q, give equivalent norms. In addition, we can replace the number 1 in 2 by any finlte constant without changing the space.

For this space the following nonhomogeneous versions of the div-curl lemma can be shown (these are special cases of Theorems 3 and 4 in [8] :

(3)

Theorem 1 ([8]). Suppose ii and ill are vector fields on lR" s bsfymg

- - 1 1 V E LP(JRn)n, WE Lq(JRn)n, 1 < p < 00, - + - = 1. P q

(a) Assume

curlW =0

in the sense of distributions. Then V . W b l ngs t th local H y spaC1! hI(JRn) with

(4) IIV . Wllhl(JRn) ~ C(lIVIILJJ(JRn) + IIf L lit \\1 L 1t •

(b) If Mnxn denotes the space of n-by-n matrices over lR a d

(5) divV = 0, curl W = B E Lq(lR",lU xn)

in the sense of distributions, then V . W belongs to th I cal Hardy spaC1! hI (JRn) with

(6) IIV . WII h' (JRn) ~ CIIVIILP(IIt") [IIWIILQ(lItn) + ~ B'J L R 1· ',J

Before continuing further, let us make clear what we mean by the divergence and curl of a vector field in the sense of distributions. Let n be an open subset of JRn, and suppose v = (VI,"" 'Un) with v. loca.lly integrable on n. For a locally integrable function f on n, one says tha.t div ii = f in the sense of distributions on n if

(7)

for all 'P E Cr(n) (i.e., smooth functions with compact support in n). Similarly, if ill = (WI, ... , w") with w, locally integrable on n , and B is an

n X n matrix of locally integrable functions B.; on n, we say curlw = B in the sense of distributions on n if (8)

NONHOMOGENEOUS DIY-CURL DECOMPOSITIONS ON A DOMAIN 1~5

for all i,j E {1, ..• ,n} and all cP E Cer(O). If the components of 11 or w are sufficiently smooth, these definitions are equivalent to the classical notions of divergence and curl via integration by parts.

Recall that C. Fefferman [9] identified the dual of the real Hardy space HI with the space BMO of functions of bounded mean oscillation, introduced by John and Nirenberg [13]. In the local case, Goldberg [11] showed that the dual of hI (JR.n) ('AD be identified with the space bmo(JR.n), the Banach space of locally integrable functions I which satisfy

(9) 1

1I/11bmo := sup III IIISI

1 II - III + sup III

I 111>1 III < 00.

I

Here I can be used to denote either balls or cubes with sides parallel to the axes, III denotes Lebesgue measure (volnme) and II is the mean of lover I, i.e., (1/111) II I. As in the case of hI, the upper-bound 1 on the size of the cubes in the definition can be replaced by any other finite nonzero constant, resulting in an equivalent norm.

In [5], the authors prove (in Theorem 2.2) a kind of dual version to the div-curl lemmas in Theorem 1, which is a local analogue of a result proved in [7J for BMO: for G E bmo(lRn),

10 .... ....

lIGllbmo ::::l sup GV· W, v,w IR"

........ where the supremum is taken over all vector fields V, W as above, satisfying (3),

.... .... with V [.P. I L and IIWIILq all bounded by 1. Here, and below, one must obviously consider only real-valued functions 9 in bmo.

Moreover, the same equation (10) holds if the vector fields, instead of (3), .... ....

satisfy 5 with B.,I1M $1 for all i,j E {l, ... ,n}, as well as IIVIILP,IIWIILQ $1. As a of these results, one is able to show (see [5, Theorem 3.1])

a decomposition of functions in h i (JRn) into nonhomogeneous div-curl quantities ........ V· W of the type found in Theorem 1, part (a) or part (b).

The goal of this paper is to prove analogues of (10) for functions in local bmo spare; on a domajn 0, a.nd obtain decomposition results for the local Hardy spaces. This was done in the case of BMO and with homogeneous, L2 div-curl quantities in 3J. and independently by Lou [16J. In [lJ homogeneous div-curl results on domains were stated nnder the assumption that one of the vector fields is a gradient, and extellded to Hardy-Sobolev spaces. Related work may be found in [12, 17J.

In the next section we introduce some definitions of Hardy spaces and BMO on domajDS, as well as explain the boundary conditions for equations (7) and (8). The statements and proofs of our results are contained in Section 3.

2. Preliminary definitions for a domain n For the moment we will just assn me 0 is an open subset of JR.n , but often we

will restrict ourselves to a Lipschitz domain, i.e., one whose boundary is made up, piecewise, of Lipschitz graphs.

Miyacbi [19] defined Hardy spaces on 0 by letting o(x) = dist(x, a~), replacing

the maximal function Min (1) by

M .. o(f)(X) = M~ 6(z) (f)(x) = sup If ... (Mx)l, "', • O<:t<:6(z)


for f E Lloc(n), and requiring Mt/>,o(f) E LI (n). The space of such functions was later denoted by Hi (n) in [6], since when the boundary is sufficiently nice (say Lipschitz), Hi(n) can be identified with the quotient space of restrictions to 0 of functions in HI (JRn

) (see [6,19)). Moreover,

IIfIlH,!(o) := IIMt/>,o(f)II£I(o) ~ inf{IIF IH1(JlR) : F fl = n· The space h~(O), corresponding to restrictions to n of functions in hI Rn ,

can be defined by replacing 5(x) = dist(x, an) in Miyachi's definition by 5 x) = min(5,dist(x,an)), for some fixed finite 5 > O. Since different choices of 5 give equivalent norms, when n is bounded one can choose 5 > diam n), so h~ n is the same as Hi(O) (with norm equivalence involving constants depending on n .

For n a Lipschitz domain, the dual of h~(n) (see [191 for the case of HI and BMO, and [2)) can be identified with the subspace

bmoz(O) = {g E bmo(JRn) : supp(g) C TI}.

Analogously, one can consider the subspace

h~(n) = {g E hI(l~n) : supp(g) C TI}.

This was originally done in [15] in the case of HI functions supported on a cl~ subset with certain geometric properties, and later in [6] for a Lipschitz d main and in [4] for a domain with smooth boundary, in connection with boundary value problems. For a bounded domain 0, H;(O) and h!(n) do not coincide since functions in H; must satisfy a vanishing moment condition over the whole domain n, while those in h! do not.

The dual of h! (0) can be identified with bmor(n , defined by requiring the supremum in (9) to be taken only over cubes 1 contained in n. In fact, one can actually require the cubes to satisfy 21 C n. This space was originally studied, in the case of BMO, by Jones [14], who showed that when the boundary of 0 is sufficiently nice, BMOr (0) is the same as the quotient space of restrictions to 0 of functions in BMO(JRn). This holds in particular when n is a Lipschitz domain, and is also true in the case of bmOr, with

Ilgllbmor(n) :::::: inf{IIG llbmo(Rn) : G n = g}.

Note that when 0 is a bounded domain, every element of BMOr(O) is in bmOr(O), but the bmor norm depends also on the norm of the function in LI(n).

Since elements of h!(O) are controlled in norm up to the boundary, when discussing div-curl quantities in this space one needs to consider the "boundary values" of the vector fields v and w. As these vector fields are only defined in V(O) and do not have traces on the boundary, the appropriate boundary conditions are expressed, as in the case of Dirichlet and Neumann boundary value problems, by specifying the class of test functions. In particular, for the equations

divv= f and curlw = B,

we now require (7) and (8) to hold in the case when the test functions do not have compact support in O. This is equivalent to saying that the vector fields

(11) V _ {v in 0 - (5 in JRn \ n

NONHOMOGENEOUS DIV-CURL DECOMPOSITIONS ON A DOMAIN 151

and

(12)

... ... satisfy div V = 1 and curl W = B in the sense of distributions on aft, with 1 and B vanishing outside of n.

When the boundary an of n is sufficiently smooth, let ii == (7'/1,"" 7'/ft) denote the outward unit normal vector. If the vector fields are sufficiently smooth (so as to have a trace on the an), we can integrate by parts in (7) and (8). If tp does not have compact support in n, the boundary values of v must satisfy ii . v = 0, and in the case of a bonnded domain, this also entails 10 1 = 0, while for w it must hold that on an

Wj7'/, = Wi7'/j,

that tV is colinear with ii. We will denote these conditions as follows. Let n be a Lipschitz domain and

suppose 1 and the components of the vector fields v and w are locally integrable on n. AB in the statement of the Nenmann problem on n, write

13 divv= 1 10 / =0 ii·v=O

in n, if n is bounded,

on an to indicate that (7) holds for all tp E C8"(JRn), and

14 curlw = B ... ... 0 nxw=

in n, on an

to indicate that (8) holds for all i,j E {I, ... , n} and all tp E C8"(JRn).

3. DiV-Cllrllemroas for local Hardy spaces and BMO on a domain

In order to prove an a.nalogue of (10) for bmo .. (n), one needs the following versions of the nonhomogeneous div-curllemma for h~(n). The first is a special case of Theorem 7 in [8]:

Theorem 2 ([8]). Suppose v and w are vector fields on an open set n Can, sahsjyutg

V- E V(n 1ll>ft) w'" E Lq (n, 1ll>n), .~ , H ~ 1 < p < 00,

and

divv = 1 E V(n), curl w = 0

1 1 -+-=1 p q ,

In 1M sense 01 distributions on n. Then v·w belongs to the local Hardy space h~(n) 1JJI,th

(15)

The below:

IIv, wllh~(n} ::; c(IIVIILP(n) + II/I1LP(o»llwIIL'(o), is a domain version of Theorem 4 in [8], whose proof we shall adapt


Theorem 3. Suppose v and ware vector fields on an open set n c JR.", satis-fying

and

divv= 0,

1 < p < 00, 1 1 -+-=1, P q

in the sense of distributions on n. Then v·w belongs to the local Hardy space h; n) with

(16) \Iv. wllhHo) ::; CllvIILP(O) (lIwIlLq(o) + LIIB., Lq 0)'

i,]

PROOF. Consider a point x E n and a cube Qi, centered at x and of sidelength 1 > 0, depending on x. We choose 1 = min(l,dist(x,an))/v'n, which guarantees Qi lies inside n. Without loss of generality assume Qi = [0, ll". Writing iJ = (VI, ••• , vn ), and fixing i, we solve - ~ Ui = Vi with mixed boundary conditions: on the two faces Xi = 0 and Xi = 1 we impose Neumann boundary values

au. _ 0 aXi - ,

and on the other faces (corresponding to Xj = 0 and x, = I, j =I- i) Dirichlet boundary values 1.1.. = O. This can be done by expanding in multiple Fourier series (with even coefficients in Xi and odd coefficients in x,, j =I- i). By the MarcinkiewiC2 multiplier theorem (see [18, Theorem 4]) the second derivatives of the solution u. will be bounded in Lo'(QI) by IIviIlLQ(Qn, for every a ::; p, . = 1, ... , n. Note that by the homogeneity of the multipliers, the constants will be independent of l. Since we have taken 1 ::; 1, we also get that \lu,lIw2.p(Qf) ::; C v.1 LP(QZ with a constant independent of l.

Set 0 = (1.1.1,""1.1.,,) and consider the function div 0 E W1,p Qf . This function satisfies

~(div 0) = -divv = 0

in the sense of distributions on Qi, since Qf C n, and moreover

~ '" au, divU = L...J- = 0 ax, on the boundary, by the choice of boundary conditions above. By the uniqueness of the solution of the Dirichlet problem in W~'P(Qf), we must have div 0 = 0 on Qi. Let A be the matrix curiO, i.e.,

aut auAij=---J. ax, ax,

These are functions in W 1 ,P(Qf) with first derivatives bounded in the LO(Qf)-norm by IIviIILQ(Qrl' for every a::; p.

Now writing A; for the jth column of the matrix A, we have, in the sense of distributions on Qf,

~ ~(a2u, a2u,) a ~ (17) div A, = ~ ~a - -a 2 = -a _ div U - ~ 1.1.; = VJ '

,-1 x, x, Xi x,


for each j = I, ..• ,n. Taking the dot product with wand recalling that we identify curl w, in the sense of distributions on n, with a matrix B whose components are in Lq(n), we have

n n

j-1 ;=1 i,j X,

n

= Ldiv(A3W j) + LAijB'jl 3=1 i<j

aga.in in the sense of distributions on Qf. Take tfI E Coo with support in B(O,1/(2y'n)) and J <fJ = I, and define, for

o < t ~ min(I,dist(x,oO)), <fJf by <fJHy) = t-n <fJ(t-1(x - y)). Since t = min (l,dist(x,aO)) .;n we have

supp(<fJn c B(x, t/2.jTi) c Qf en. - .. . -

Denote B(x, t 2y'n) by Bt. We integrate iJ· w aga.inst tflf I noting that equation (17) holds even if we change -A, by adding a vector field which is constant on Qf. In particular we modify each .. - ..

A.J by subtracting its average (Aij)Bz over Bt. Integration by parts yields: •

• &., Yi •

+L i<j

Take ct, {:J with 1 < 0: < p, 1 < f3 < q and 1/0: + 1/ f3 = 1 + l/n. The SobolevPoincare inequality in 13:, together with the fact that t ~ 1, gives (see the proof of Lemma ILl in [7]):

1/~

<Pt * (ii· w)(x)1 ~ CjV<fJl1oo L • • ~,3

+ CII<fJlloo L t-'" _IVAij I'" en _IBij I~ i,j Bi Br

1/~

~ C",M(liJI"')(x)1/et M(lwl~)(X)l/~ + LM(IBijl~)(x)l/~ . • • 1,3

Here the Hardy Littlewood maximal function on JR"', denoted by M, is applied to the functions I VIet ,lwl.8 and IBijl.B by extending them by zero outside n. The constant depends on the choice of <fJ but not on t or x .

. g that the ma.ximal function is bOlInded on U(JR"'), r > 1, we conclude that:

160 D.-C CHANG ET AL.

1 sup \4>t'" (11. W)(X)\ dx o O<t<dlst(:t,BO)

~ C",(fn (M(\ii'\Q) (X))p/Q dx) 1/1'

X [(fn (M(\w\~)(X))q/~ dx) l/q + ~ (10 (M( Bo) tJ)(X))q fJ dx) 1 1 ~ C",IIii'IILP(O) [lI w \\LQ(O) + EIIB,} IILO(O)]'

This shows ii'. W E h~(n), and (16) holds. o

Lemma 4. Suppose ii' and ware vector fields on a Lipschitz doma nne Rn, satisfying the hypotheses of either Theorem 2 or Theorem 3, but with the cond t ona on the divergence and the curl satisfied in the stronger sense of 13 and 14. Then ii'. w E h!(n) with norm bounded as in (15) or (16).

PROOF. Given such vector fields v and won il, define the zero extensi ns V and W as in (ll) and (12). The V and Lq norms of V and l-V are the same as those of ii' and won n. Moreover, conditions (13) and (14) guarantee that V and lV satisfy (3) (respectively (5)) in the sense of distributions on R". Therefore, by using Theorem 1, part (a) (respectively part (b)), we can conclude that V· lV E hl(Rn with the appropriate bound on its norm. But V . W is equal to zero outside n and is ii'. won n, hence this is a function in h!(n). The h! norm is the same as the h1

norm and the bounds can be given in terms of the V and Lq norms of the relevant quantities on n. 0

Now we can proceed to state and prove the local bmo versions of the div-curl lemma on a Lipschitz domain:

Theorem 5. Let n c IRn be a Lipschitz domain.

(a) If 9 E bmoz(n), then

(18) I\gl\bmoa ~ sup 1 9 ii . W, V,w 0

where the supremum is taken over all vector fields ii E V(n,IRn), wE Lq(n,R.n ),

1lii'lIv(o) ~ 1, IIw\\LQ(O) ~ 1, satisfying (3) in the sense of distributions on n, with IIf\\v(o) ~ 1.

(b) If 9 E bmoz(n), then equation (18) holds with the supremum now taken over all vector fields ii E Lp(n,IRn), w E Lq(n,Rn ), llii'\ LP(O) ::; 1, \wIlL.(O) ~ 1, satisfying (5) in the sense of distributions on n, with liB,} ilL. (n) ~ 1 for all i,j E 1, ... ,n.

(e) If 9 E bmo,.(n) then

IIgllbmor ~SUPl gii'.w, v,w 0

the supremum being taken over aU vector fields ii' and til as in part (a) or in part (b), but satisfying the stronger conditions (13) and (14).

NONHOMOGENEOUS DIV-CURL DEOOMPOSITIONS ON A DOMAIN 161

PROOF. Let 9 E bmo.z:(n) (real-valued) and take v,'Iii as in the hypotheses of part (a) (respectively part (b)). By Theorem 2 (resp. Theorem 3), the dot product ii. w belongs to h~(n) with norm bounded by a constant. The duality of bmo.z:(O) with h~(O) then gives

9 v . w =:; ClIgllbmo.' o

Conversely, by the nature of bmo .. (O), the extension G of 9 to IRn by setting it zero outside n is in BMO(lRn) with \\Gllbmo ~ I\gllbmo.' Hence, by (10), one has

IIgl\bmo. ~ sup G V . W = sup 9 Vl o . Wlo' v,w IR" V.W 0

~ ~

where the supremum is taken over all vector fields V E.LP(JRn, lRn), W ELq(lRn ,lRn), V £p =:; 1, lV \L' =:; 1, satisfying (3) (resp. (5)) in the sense of distributions on

R". The restrictions v = Vlo' w = Wlo satisfy the same conditions in 0, proving the inequality ;S in (18).

IT 9 E bmor(n) and v, ware as in part (c), by Lemma 4 ii· 'Iii E h;(O) with norm bounded by a consta.nt, so the duality of bmor and h; implies

This shows that

sup gii· w =:; C\\g\\bmo~. V,W 0

It rema.ins to prove the other direction, i.e.,

IIgllbmo~ =:; C'sup ;1,10

The left-hand side is given by

o ~ ~

gv· w.

1 1 sup Q\ \g(x) - gQ\ dx + sup \Q\ \g(x)\ dx. QCO Q QCO Q Q 9 IQI>l

AB explajned in the proof of Theorem 2.1 in [3] (for the case of BMOr(O) but the same arguments apply to bmor (0)), it suffices to take the supremum over cubes Q satisfying Q = 2Q C n (or with some constant Co replacing 2). In that case it just remains to point out that in the proof of estimate (10) in [5] (see the proof of Theorem 2.2., Case I), it was shown that for a. ball B c IRn with radius bounded by 1,

1

IBI Ig(x) - 9BI2 dx B

1/2

=:;Cnsup gil· w,

where the supremum is taken over all vector fields il, wE C(f(B) with \\ii\lLP ::5 1, -tV L ::5 1 and div v = 0, curl w = O. There we took B = 2B but the argument - -immediately applies to B = CoB for some Co > 1. Note that if B c 0, such vector fields will yanish on the boundary ao and thus satisfy the boundary conditions (13) and (14).


Similarly, for a ball B c IRn with radius larger than 1, we showed in [51 (see the proof of Theorem 2.2., Case I) that

C~I fB 1g(x)12 dxy/2 ::; Gnsup f gv·w,

where this time the supremum can be taken over all vector fields V, w E Gil" B with IIvilLp ::; 1, IIwl0 ::; 1 satisfying the relaxed div-curl conditions (3), or alternatively the supremum can be taken over such vector fields satisfying 5. Again such vector fields will automatically satisfy (13) and (14)-the boundary conditions follow from the vanishing on the boundary and the condition 10 div 11 = 0, in the case

of bounded n, follows from the divergence theorem since we are now dealing 'with smooth functions. 0

Finally we arrive at the desired nonhomogeneous div-curl decompositions for the local Hardy spaces on n. These are corollaries of Theorem 5 and follow from the duality between bmoz and h~ (respectively bmor and h~ by using Lemmas m.1 and III.2 in [71:

Theorem 6. Let n c lRn be a Lipschitz domain and 1 < p < 00, 1 p+ 1 q = 1.

(a) For a function f in h~(n), there exists a seque ce of scal TS {A,,} th. I:~=1IA" 1< 00, and sequences of vector fields {v,,} in lJ' n, 1R {w,,} m Lq n, Rn with \Iv" \I Lp, \lw,,10 ::; 1 for all k, satisfying, for each k, rend t n 3 m the sense of distributions on n, so that

00

f = '2: A"v" ·w". "=1

(b) The same result holds as in part (a) but with V" and w" sahsfymg 5 instead of (3), for each k.

(c) For a function f E h;(n), there exists a sequence oj scalars {A"} 'lVIth I:~=1IAkl < 00, and sequences of vector fields {Vk} and {Wk}, as In part (a or part (b), but satisfying the div-curl condltions in the stronger sense oj 13 for each Vk and (14) for each Wk, so that

00

f = '2: AkVI.: ·'Iiik. 1.:-1

Remark. As pointed out in Section 2, when the domain n is bounded the "local" Hardy space h~(n) coincides with H;(n) and similarly for BMOz(n) and bmoz(n). By taking the constants in the definitions and proofs sufficiently large (depending on the size of n), we do not need to deal with the case of "large" balls or cubes, 80 everything reverts to the homogeneous case . .Ai; previously mentioned, this case was dealt with in [31 and (16), but only for p = q = 2, so the current results are a generalization of the older ones.

References

1. P. Auscher, E. Russ, a.nd P. Tchamitchia.n, Hardy Sobolev spaces on strongly LIpschItz domaIns olRn, J. Fund. Anal. 218 (2005), no. 1,54 109.

2. D.-C. Cha.ng, The dual 01 Hardy spaces on Q bounded domain In Rn, Forum Math. 6 (1994), no. I, 65 81.


3. D.-C. Chang, G. Dafni, and C. Sadosky, A dill-curl lemma in BMO on a domain, Ha.rmonic analysis, signal and complexity, Progr. Math., vol. 238, Birkhiuser, Boston, MA, 2005, pp. 55 65.

4. D.-C. Chang, G. Dafni, and E. M. Stein, Hardy space8, BMO, and bounda", value probleml for the Laplacian 0 .. a smooth domain In R", '!tans. Amer. Math. Soc. 351 (1999), no. 4, 1605 1661.

5. D.-C. CbaDg, G. Dafni, and Hong Vue, A dill-curl decomposlnon for the local Hardy space, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3369 3377.

6. D.-C. Chang, S. G. Krantz, and E. M. Stein, HI' thea", on a smooth domain in RN and elllpt\c boundary "alue problems, J. Funct. Anal. 114 (1993), no. 2, 286 347.

7 R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactne8s and Hardy spaces, J. Math. Appl. (9) 72 (1993), no. 3, 247 286.

8. G. Dafni, Nonhomogeneous diu-curl lemmas and local Hardy spaces, Adv. Differential Equations 10 (2005), no. 5, 505 526.

9. C. Fefferman, Characte"zo,nons of bounded mean oscil/ation, Bull. Amer. Math. Soc. 77 1971), 587 588.

10 C. Fefferman and E. M. Stein, HI' spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137 193.

11.0 Goldberg, A tleiMOn of real Hardy spaCe8, Duke Math. J. 46 (1979), no. I, 27-42. 12. J. Hogan, C. Li, A. McIntosh, and K. Zhang, Global higher Inte9l"Qbility of Jacobians on

bounded domains, Ann. Inst. H. Poincare Anal. Non Lineaire 17 (2000), no. 2, 193-217. 13. F John and L. Nirenberg, On /unctions of bounded mean oscillation, Comm. Pure Appl.

Math 14 1961),415 426. 14. P W. Jones, &tension theorems for BMO, Indiana Univ. Math. J. 29 (1980), no. 1,41-66. 15 A P. Sjogren, and H. Wallin, Hardy and Lipschitz spaces on subsets ofRR, Studia

Math. 80 1984), no 2, 141-166. 16 Z Lou, Dau-curl type theorems on Lipschitz domains, Bull. Austral. Math. Soc. 72 (2005),

no 1,31-38. 17 Z. Lou and A. McIntosh, Hardy spaCe8 of exact f011l18 on Lipschitz domains in IRN , Indiana

Univ. Math. J 53 2004), no. 2, 583-611. 18. J M81c;inkiewicz, Sur les multlpllcateurs des serie8 de Fourier, Studia Math. 8 (1939),78-91. 19 A. Mlyachi, lP' space8 otler open subsets ofRR, Studia Math. 95 (1990), no. 3, 205-228.

OF MATHEMATICS, GEORGETOWN UNIVERSITY, WASHINGTON DC, 20057, USA E-mad address: chang«lmath. georgetown. edu

DEPARTMENT OF MATHEMATICS AND STATISTICS, CONCORDIA UNIVERSITY, 1455 DE MAISON"'EUVE BIND. W, MONTREAL, QC H3G IM8, CANADA

E-mml address: gdafnibathstat. concordia. ca

DEPARTMENT OF MATHEMATICS AND STATISTICS, CONCORDIA UNIVERSITY, 1455 DE MAISON"'EUVE BLVD. W, MONTREAL, QC H3G IM8, CANADA

Current address: Department of Mathematics and Informatics, Trine University, Angola, IN 46703 USA

&mall aAAress: yueh«ltrine. edu

Oontre d. Recherch ... M .. tMrn .. tlqu"" CRM Proceeding" and Lecture Note. Voluble Ill, 3010

On the Bohr Radius for Simply Connected Plane Domains

Richard Fournier and Stephan Ruscheweyh

ABSTRACT. We give various estimates and discllss sharpness questions for a generalized Bohr radius applicable to simply connected domains of the complex plane.

1. Introduction and statement of the results

Let II) = {z Ilzl < I} be the open unit disc in the complex plane and H(II)) the class of analytic maps on lIll. Let J E H(II)) with !(z) = 2::'0 alczlc and !(]I))) ~ D. Then,

00

~)alclrlc ~ 1 if r ~ ~. 1=0

This was first obtained by Harald Bohr [2] in 1914 with the constant ~ replaced by ~ and later improved by M. Riesz and others who established that in this context the constant ~ is best possible; different proofs were later published while similar problems were considered for Hp spaces or more abstract spaces or else in the context of several complex variables by a number of authors. We refer to a paper [6] and a book [7] by Kresin and Maz'ya for a rather complete survey of recent and less recent related results.

Our point of view is the following: we consider a simply connected domain V with II) ~ V and define the Bohr constant B = BD 88

00 00

sup r E (0,1) : ~)alclrlc ~ 1 for all J(z) := ~ alczk E 8(V), z E ]I)) Ic=O 1c=0

while 8(V) is the class of functions! E H(V) such that f(V) ~ D. Clearly Ell) = 1 coincides with the classical Bohr radius and we wish to estimate ED for more general domains V. Our main results are the following:

2000 M Subject Cltusification. Primary 30B10; Secondary 30A10. Key wo.,u and phraau. Bohr radius, power series, plane simply connected domains. This is the final fOlm ofthe paper.

@)2010 American Mathematical Society

166

166 R. FOURNIER AND ST. RUSCHEWEYH

Theorem 1. Let 0::; 'Y < 1 and V.., the disc {W: \w+'Y/(I-'Y)\ < 1/(I-'Y)}. Then Bp.., = p.., := (1 + 'Y)/(3 + 'Y) and L:~=ola" 14 = 1 holds Jor a funcbon J(z) = L:~=oa"z" in 8('0..,) iJ and only iJ J == c with Ie = 1.

Theorem 2. Let V be a simply connected domain with V ;2 III and let

>. := >'('0):= sup {I latl 12 : ao t:. J(z) = f: a"z", ZED}. /EB(P) - ao "=0

"<!1

Then 1/(1 + 2>.) ::; Bp and the equality L:~=ola" (1/(1 + 2>') " = 1 holds Jor a function J(z) = 2:~=0 a"z" in 8(V) iJ and only iJ J == e 'IU&th e = 1.

In the case of a convex domain V, it is possible to estimate Bu in terms of the conformal radius:

Theorem 3. Let V be a convex domain with '0;2 D and F z):= A1z + 0 z a conJormal map oJ'O onto II]) with Al > O. Then

.B:= min (1, 4~J ::; Bu·

Our next result is a limiting case of Theorem 1, but should also be compared to Theorem 3.

Corollary 4. Let P denote the half-plane {z \ Re z < I} and f z = 2:~=0 anzn for Izl < 1 where f E 8(P). Then

(1) if 0 ::; T ::; }.

2. Proofs

The proof of Theorem 1 is based on the following result which may be of independent interest:

Lemma 5. Let a E II]) and J E 8(1I]) with 00

J(z) = L o,,(z - a)\ Iz - a ::; 1 - a. "=0

Then

(2) 00

Llo" Irk::; 1 k=o

. l-lal2

if r ::; TO:= 3 + lal .

F'u.rthermore, ro is the largest number with this property, and 00

Llo"lr~ = 1 "=0

if and only if J == e with lei = 1.

PROOF. We make use of the following estimate (see [9] for a proof; this estimate is of course an extension of the Schwarz-Pick inequality and has shown to be useful in a number of situations):

( ) I I ( I 1)"-1 1 - 10 012

3 0" ::; 1 + a (1 _ laI2 )" ' k ~ 1,

ON THE BOHR RADIUS FOR SIMPLY CONNECTED PLANE DOMAINS 161

and note that lao I ~ 1. We get

00 Ie < (1 - laol2)r 10-0

and the last quantity is seen to be less or equal to 1 for all admissible ao if and only if 2r/(1 + lal)(I-lal- r) ~ 1. This leads to (2).

Again using (3) it is easily established that 00

(4) Llalelr~ = 1 -< :. (Iaol = 1 and ale = 0, k ~ 1). 10=0

Further, the functions 00

z-b '" Ib(Z) = 1 _ b := L..t alo(a, b)(z - a)\ z 10=0

belong to B(IDl) and one has 00

5) Llalo(a, b)lrlo ~ 1 for all bE lIll 10=0

b E lIll,

if and only if r ::; ro which implies that the constant ro is indeed optimal with to the statement of the lemma. 0

PROOF OF THEOREM 1. For some 0 ~, ~ 1 let 1 E B(V-y) such that I(z) = E:"=oalez"', z E rr». Then g(z) := I(z - ,)/(1 - ,») belongs to B(lIll) and if Z -, ~ 1 - ,1, we have

g(z) = 1

The lemma now gives,

10=0

z-, 1-,

P 1-,

10

<1 -

and l-r)/(3+-y) ~ Bv.., follows together with the statement of equality which is a of (4). The functions 1 E B(V-y) defined by 1(z-,)/(I-,» = Ib(Z),

b, z E rr», may be used as in (5) to show that the constant (1 + ,)/(3 +,) is best possible with respect to Theorem 1, i.e. Bv.., = (1 + ,)/(3 + ,). 0

PROOF OF COROLLARY 4. Under the hypothesis, 1 E B(V-y) for all, in [0,1) and the result follows from Theorem 1 by letting , -+ 1. The constant ~ is best possible as can be seen from the Taylor expansion at the origin of the functions

z/(z - 2) - a g .. (z) = 1- az/(z _ 2)' Re(z) < 1, a E lIll.

We omit the details. Due to the limiting process used in the above argument, the case of equality

in (1) is no more a simple consequence of (4) and a separate proof must be given. We first apply inequality (3) to obtain


for any 9 E 8('0"1)' g(z) = E~ OOlcZIr, zED. This leads to

lalcl 5 ~(I-laoI2), k ~ 1

for any f E 8(P) ::= nO$"1<18('D"1) such that fez) = E:.o alczlr, zED. If, in particular, E~=olQ,/cI(~)1c = 1 for f as above, we obtain

1 00 (1)1r 1 ao 2

1 $ laol + 2(1-lao I2) t; 2 = lao + - 2

and laol = 1, Le., f is a constant function of modulus 1. 0

PROOF OF THEOREM 2. The proof of the main statement is rather straight-forward: if f belongs to 8('0) with fez) = E:'o a/czlc in D, then for any rEI

~ Ic 2 >.r L.",lalclr ~ laol + (1 - ao ) 1 _ r Ic=O

so that E~=olalclrlc ~ 1 when 2>.r/(I-r) 5 1, i.e., when r ~ 1 1+2 . This early shows that Bp ~ 1/(1 + 2>'). As in the last step of the proof f Cor llary 4, the equality E%"=olalcl(I/1 + 2>.)1c = 1 holds only for constant functions f modulus 1.

o Remark. It does not seem entirely trivial to characterize domains V for v. hich

Bp = 1/(1+2>'). It is not hard to see that this indeed is the case see 6 f r a hInt in this direction) when '0 = '0"1' 0 ~ "( < 1. Next we exhibit a class f d mains f r which this is definitely not true. Consider a sufficiently regular simply connected domain '0 2 D such that {)1) n 8[Jl i= 0, together with the ass ci ted conformal map F of '0 onto [Jl with F(O) = 0, F'(O) =: Al > O. The inverse m p F-l: D -+ '0 satisfies F-l(u) = (I/Al)u + o(u) and by the classical growth theorem,

(7) A1IF-1(U)1 ~ (lll~\)2' u ED.

Our hypothesis on 8[Jl and 8[Jl implies that for some sequence of elements uJ E D we have IUjl-+ 1 and IF-1(uJ )I-+ 1. It then follows from (7) that Al ~ !.

Let fez) = '2::'0 alczlc I zED, for some J in B('O) where V is to be determined later. We have, thanks to estimates due to Avkhadiev andWirths [1, pp.60 75},

4n - 1/ 2

lanl $ ../n + 1 Ai(1 - laoI2), n ~ 2.

This estimate is actually also valid for n = 1: For if w(u) := J(~l(U)), U ED, then w E 8([Jl) and ao = w(O), a1 = w'(O)Al and

lall Iw'(O)1 rn 1-laol2 = 1_lw(0)12A1 $ Al < v2Al .

Therefore 00 00 4n-l 2 An 'Llanlrn 5 laol + (1-laoI2

) L ,;nnl rn $ 1 nO n~ +

if E:'_1(4A l r)"/"/n + 151. Let X be the unique root in (0,1) of the equation

00 xn 'L J::"'71 = l. n-1 v n + 1

ON THE BOHR RADIUS FOR SIMPLY CONNECTED PLANE DOMAINS 169

We now produce domains 'D for which 4A1/(1 + 2A) < X: this will clearly imply that for such domains 1/(1 + 2A) < E'D.

Since Al = F'(O) = 1F'(0)\/(1 - \F(O)l2) < A, it shall be sufficient to identify domains 'D ;2 D with 4Al/(1 + 2A1) < X, i.e., Al < 1/(4 - 2X). Since ~ < 1/(4 - 2X) < ~, it follows that any simply connected domain V containing the plane P1,0 (for which Al = !) and close enough, in the sense of kernel convergence, to the slit plane C \ [1,00) (for which Al = ~) will serve as an example.

PROOF OF THEOREM 3. Let 'D be a convex domain, V ~ JI)) and F the conformal map as above with F(O) = 0, F'(O) =: Al > O.

IT f E B(V) and fez) = E;:'o akzk, Z E JI)), another estimate due to Avkhadiev and Wirths [1, pp. 60-76] yields

(8) lakl ~ 2k- 1 A~(l -laoI2), k 2: 2,

and as in the proof of Theorem 2, this extends to k = 1. First assume that Al ~ ~j then by (8),

00 1200

k=O k=l

and this last quantity is easily seen to be less or equal to 1 for all admissible ao. When Al > ! and r ~ 1/(4A1 ), we also obtain from (8)

00 2 00

k=O k=l

_0 2 2 0 2-k=l

and the result follows. It should also be clear from our arguments that the equality case E~=o ak l.Bk = 1 can occur only when f is a constant function of modulus ona 0

Remarks. (1) When Al > !, it again does not seem easy to characterize the convex domajns 'D for which .B = 1/(4A1) = Bv. Indeed, 1/(4A1) < Bv = l if V is the nnit disc ID> and 1/(4A1 ) = Bv = 1/(1 + 2A) = ~ if V is the half-plane P. Further, we sometimes have 1/(4A1 ) < 1/(1 + 2A) (as in the case of the unit disc) and 1/(4A1) > 1/(1 + 2A) (as in the case where V is a disc centered at the origin with radius> 4).

(2) Theorem 3 does not necessarily hold for non-convex domains. Let V be the slit plane C\(-oo,-l]j then F(z) is the inverse of 4k(z) where k(z) := z/(l- Z)2 is the Koebe map and Al = ! with F(z) = ~z - ~z2 + ;2z3 + .... For 0 < a < 1 define f E B(V) by

F(z) + a. fez) = 1 + aF(z)

(1- a2) (1- a2)(a + 2) 2 (1 - a2)(a2 + 4a + 10) 3 = a - , 4 z + 16 z + '64 z + ...

where zED. Then, for a = .9, we have

1- a2 (1 - a2)(a + 2) + (1- a2)(a2 + 4a + 10) _ 1 0247 > 1

a + 4 + 16 64 -.. . . ,


which shows that Bv < 1 = {3.

3. Conclusion

Given a domain V ~ lD>, the determination of the function 00

(9) M(r, V) := sup ~)anlrn, o ~ r < 1, n=O

(here the sup is taken over all functions I(z) = 1::'=0 anzn , zED, in B V)) may be seen as a generalized Bohr problem. Because the coefficients an of functions I in B(V) are uniformly bounded, it should be clear that M(r) = 1 for r sufficiently small and indeed M(r) = 1 for 0 ~ r ~ Bv where of course ~ ~ Bv.

Very little seems to be known about the function M(r, V) in general; a result of Bombieri [3] (see also [8] for related matters) says that M(r,lD» = (3-v'8 1- r2 r when l :::; r :::; 1/v'2 (Bombieri studies in fact the inverse function of M T,lD> • It also follows from the results of Bombieri that M(r, lD» :::; 1/.../1 - r2 if 0 < r < 1 and a recent result due to Bombieri and Bourgain [4] says that M r,lD» < 1 .../1- r2 if 1/v'2 < r < 1; in [4], a deeper result implies that 1/.../1- r2 is in some sense the sharp order of growth of M(r, lD» as r ~ 1.

Think of B(V) as a topological vector space endowed Wlth the topology of uniform convergence on compact subsets of V. It follows from a simple compacity argument that the sup in (9) is indeed a maximum and there exists for each rEO, 1 a function Ir(z) := 1::=0 an(fr)z" such that M(r,lD» = 1::=0 an Ir rn. Bombieri has proved in [3] that Ir is a disc automorphism (Le., a Blaschke product of order 1) when ~ < r :::; 1/,;2.

This last result can be extended to general domains V in the following fashion; let, given r E (0,1), Ir(z) = 1::=0 an(fr)zn with an(fr) = an Ir e!9.,. where 6n = 6n (r) is an angle in [0,271"). We define a linear functional L over B V) by

00 00

if/(z) = Lanzn, z <1. n=O n=O

The complex-valued functional L is continuous over B(V) and Re L is not constant there. Further

00 00

Re L(f) = Re L an(f)e-i9"rn :::; L\an(f)\rn

n=O n=O 00

:::; ~:)an(fr)lrn = L(fr) = ReL(fr). n=O

Since I E B(V) ~ I = W 0 F

where we B(lD» and F is a conformal map of V onto lD>, we may therefore think of L as a continuous linear functional over B(lD» whose real part is nonconstant there and maximized by a function Wr E B(lD» with

Ir(z) = wr(F(z»), % E V.

It is well-known [5, Chapter 4] that such a function Wr must be a finite Blaschke product. We may therefore state that any function I for which the sup is attained in (9) is such that I 0 F-1 is a finite Blaschke product.

ON THE BOHR RADIUS FOR SIMPLY OONNECTED PLANE DOMAINS 171

References

1. F. G. Avkhadiev and K.-J. Wirths, Schwarz fuk type Inequa"t\e., Front. Math., Blrkhil.user, Basel, 2009.

2. H. Bohr, A theorem power se"ea, Proc. London Math. Soc. 13 (1914), 1 Ii. s. E. Bombieri, Sopro un teorerno cit H. Bohr e G. RICCI .ulle funAOfta mB991oront\ delle .ene d,

poten.te, Boll. Un. Mat. Ita!. (3) 17 (1962), 276 - 282. 4. E. Bombieri and J. Bourgain, A rernork on Bohr', inequality, Int. Math. Res. Not. 80 (2004),

4307-4330. S. D J Hallenbeck and T. H. MacGregor, Linear problema and oonlleXlty techniques In geometnc

/uncbo" theory, Monogr. Stud. Math., vol. 22, Pitman, Boston, MA, 1984. 6. G. and V Maz ya, SM.rp Bohr type part estimates, Comput. Methods FUnct. Theory

or 20(7), DO 1, 151 165. T , ShGrp ,ro/-port theal'ellis: A unljied approach, Lecture Notes in Math., vol. 1903, S~, ~lin, 2007

S. G RIca, Complement1 a un teorerno dl H. Bohr nguamante Ie sene dl potenze, Rev. Un. Mat. Argentma IT (1955), 185 195.

9 St. Ruscheweyh, Two on bounded analytic functions, Serdica 11 (1985), no. 2, 200-202.

CENTRE DE RECHERCHES MATHEMATIQUES, UNlVERSITE DE MONTREAL, C.P, 6128, SUCC,

CEImtE-\'l1J.B, MONTREAl" QC H3C 3J7, CANADA E-mail oddress:fouruier&dma.nmontreal.ca

}. THENATISCHES INS'I'i'fUT, UNlVERSITAT WURZBURG, 97074 WURZBURG, GERMANY

E-mcill address: ruscheweyhlbathematik .nni-vuerzburg. de

O .. nt_ de R&eh"1'eheA M&th4!lmatlquel CRM Proe •• dln~. and Le<:ture Not"" Volumo Ill, 3010

Completeness of the System {J(AnZ)} in L![O]

Andre Boivin and Changzhong Zhu

ABSTRACT. Given an analytic function I defined on the Riemann surface of the logarithm and represented by a generalized power series with complex exponents {'TIc}, a sequence of complex numbers {An}, and an unbounded dom"in n on the complex .c plane, we study the completeness of the system {/ AnZ}} in L~[nl (mean square approximation).

1. Introduction

For aD entire function I(z) and a sequence {An} of complex numbers, the completeness in a domain n of the system {/(AnZ)} in £I'-norms has been studied by many authors under various conditions on I, {An}, p and n. See, for example, 1,9; 10; 12, Chapter 4; 14-16].

In particular, M. M. Dzhrbasian studied the completeness of {/(AnZ)} in L2 when n is an unbounded domain which does not contain the origin and is located utside aD angle with vertex at the origin (see [6, Section 5]). In [3], we also

considered this problem. We gave some sufficient conditions under which the system {f A Z } is complete. Our conditions are different, though similar, to Dzhrbasian's.

results are recalled in Section 2. Besides, in [3, Section 3], we also studied a SImilar question where this time, the entire function I is replaced by one analytic

n the Riemann surface of the logarithm. This generalized a result obtained by X. Shen in [22]. We used in [3] the classical Ritt order and Ritt type to characterize the growth of the function. This seems not suitable in general, as we pointed out m a later paper [5]. Hence, we introduced a modified Ritt order and modified Ritt type in [5]. After recalling the definitions, we will use them in this paper. Corresponding to this cha.nge, we modify related arguments used in (the second part of [3]. Besides, comparing with [3], in this paper we will impose weaker conditions on the sequence of exponents {Tk}. In particular, we do not assume that the limit limk-+oo k/ITkl exists.

The paper is organized as follows. In the next two sections, we review some of the definitions, notations, examples and results from [3-5]. In Sections 4 and 5,

2000 Mathematt.C8 Subject Cla88ijicatlon. 30E10,30B60. Kell won18 and phra8e8. mean square approximation, completeness, analytic functions, un

bounded domam. Research supporfP.d by NSERC (Canada). 1'hls 18 the final form of the paper.

173

@2010 American Mathem"Iic,,! Sooiety


we present three new results: An estimate of the coefficients in the (generalized) Dirichlet series of an entire function in terms of the modified Ritt order and type; a uniqueness theorem for analytic functions defined on the Riemann surface of the logarithm; and finally conditions for the completeness of the system {ZTk} in L~[nl, where {Tk} is a sequence of complex numbers. In Section 6, we give our main result on the completeness of the system f(A .. z)} in L![nj. The proofs of a few leIIlIIlaS are gathered in the last section.

2. Dzhrbasian's theorem

Let 0 be a domain in the complex z-plane. Denote by L![Oj the space of functions 9 analytic on 0 which satisfy

1 i lg (z)\2 dxdy < 00 (z = x + iy).

Endowed with the inner product

(g, h) = 110 g(z)h(z) dxdy,

and associated norm IIgll = (g,g)1/2, L![O] becomes a Hilbert space. A sequence {h .. } (h .. E L![O}, n = 1,2, ... ) is complete in L![O] if for any 9 E L![Oj and any c > 0, there is a finite linear combination h of elements of the sequence {h.,.}, such that IIg - hll < c.

For a function fez) and a sequence {An} of complex numbers we studied in [3] the completeness of {J(A .. Z)} in L![O].

For f (z), we assumed that it is an entire function with p wer series expansion 0Cl

(2.1) f(z) = Lakzk, ak=lO(k=O,1,2, ... k=O

A simple example of such an entire function is fez) = eZ• Hence, our results

provided sufficient conditions for the completeness of the system {e~ Z} in L![O]. For {A .. }, we assumed it to be a sequence of complex numbers with

(2.2) An =I O.

For other conditions on {An}, Sl'e Theorem 2.1 below. For 0, we assumed it is an unbounded simply-connected domain satisfying the

following two conditions:

Condition 0(1). For r > 0, let u(r) denote the Lebesgue measure of the intersection of the circle Izl = r and OJ we assume that there e:nsts ro > 0 such that for r > ro,

where p( r) has the form

(2.3)

u(r) ~ exp( -p(r)) ,

per} = o/rlf',

for two positive constants 0/ and s' .

Condition O(Il). The complement of 0 consists of m unbounded simply connected domains G, (i = 1,2, ... , m), each Gi contains an angle domain A, with measure 1r / Qi, 01, > ~ (see Figure 1).

COMPLETENESS OF THE SYSTEM {/(~".t:)} IN £!\O\ 175

y

o x

FIGURE 1. Domain 0

Remark 1. Since ol and 8 are positive, it follows from condition 0(1) that there exists a. constant C, 0 < C < 00, such that

1'0 00 00

du< - 21rrdr + u(r)dr~G+ o 0 1'0 1'0

, -a'r-e dr < +00.

'Ib study the completeness of {J(>'nz)} in L~[O], we needed a result on the completeness of the sequence {I, z, z2, z3, .•. } in L~[Ol, which is a special case of a

of M. M. Dzhrbasian (see [6], or [19, Theorem 10.1]): Let us define {J by

2.4 rJ = max{ 01, •.. , Om},

where the a. are the constants appearing in condition O(U).

Theorem (M. M. Dzhrbasian). Suppose that 0 is a domain satisfying condmons 0 I) and O(II) and that 8' and rJ are defined by (2.3) and (2.4), respectively. If

1 r 1+6 - s' dr = +00,

00

where foo means that the lower limit of the integral is a sufficiently large number, then the system {I, z, z2, Z3, ... } is complete in L~ [OJ.

Remark 2. The "sufficient large number" in the integral condition above can be replaced by "strictly positive number." In fact, the integral condition can be replaced. by the simpler condition iJ ~ s'.

Example 1. This example is taken from [31. Let 0 be the unbounded domain oontaining the real axis and having the curves y = ±ke-OI:~ (-00 < x < 00) as its boundary. It is not hard to see that 0 satisfies conditions 0(1) and O(II) with p r) = ~,-2 (i.e., s' = 2 and eX = !) and m = 2, 01 = 02 = 1. So we have rJ = 1, and

00 dr 00

dr = +00.

Let us now recall some concepts from the theory of entire functions (see, for example, 112, Chapters 1 and 4]). Let ¢ be an entire function. The quantity

. log log M",(r) p=limsup" '

1'-+00 log r


is called the order of </1, and if 0 < p < 00, the quantity

1. log M",(r)

(f = lmsup r-+oo rP

is called the type of </1, where

M",(r) = max\</1(z)l. \zl=r

For a sequence {An} of complex numbers, we denote by nCr the number (counted according to multiplicities) of An with I>"" < r.

Now we state the first main result from [31.

Theorem 2.1. Assume that f, {An} and n sat~sfy cond~tions 2.1 , 2.2 ,n I , and n(Il) stated previously in this section. Let p and u be the order and type of the entire function fez), respectively, and assume that 0 e - (pu/ fJ ,

r-+oo rB pfj s' 0'

or

(2.5') liminf n~r) > (~)pfj (pa)B fj, r-+oo rS p{3 s' 0'

where f3 = l/(s' - p) > O. If

JOO rl:;-B' = +00

where {) is defined in (2.4), then the system {J(AnZ)}, n = 1,2,3, .. 18 complete in L~[n1.

Example 2 (Taken from (31). Let n be the domain described in Example 1 and let fez) = eZ

• Then f is an entire function with p = u = 1. It thus follows that f3 = l/(s' - p) = 1 and s' pf3 = 2. Consequently (2.5) and (2.5 become

and

(2.5')*

. nCr) lim sup -2- > 2e,

r-+oo r

I, 'nf nCr) 2 lml -2- > , r-+oo r

respectively. Hence by Theorem 2.1, if {An} satisfies (2) and (2.5)* or (2.5')* (for example An = In/3ei), then the system {eA .... }~l' is complete in L![n].

3. The Riemann surface of the logarithm

For the remaining of this paper, we assume that! is an analytic function defined on the Riemann surface of the logarithm and is represented by a generalized power series

(3.1) 00

fez) = LdkZT~, k 1

z == rei8 (r > 0, \9\ < 00),

COMPLETENESS OF 'l'HE SYSTEM {/(),n%)} IN L! 01 177

or, which is the same, we assume that F(s) - f(e ') is an entire function represented by the (generalized) Dirichlet series

00

(3.2) F(s) = 'E dke 'I'k8

, 8 = U + iv k 1

where {nJ is a sequence of complex numbers. In the sections that follow, we will study the completeness of the system

{j(AnZ)} in L~[nl. X. Shen studied this problem for the case when {Tk} is a sequence of real numbers (see [22]). We consider the case when the Tk are complex.

We now make some assumptions on the exponential sequence {Tk}, the domain n and the sequence {An}.

For the sequence of complex numbers {Tk}, we will assume some or all of the following (for (II)(3.5)), see [21]; for (III)' and (III)", see [23]):

3.3)

(I) 0 < ITll < IT21 < ... < 1,-,.\ < "'; (II)

1· k DO< lmsup I 1= k-+oo ,-,.

(0 < DO< < 00),

3.4) (Do< > 0),

and n ... (r) - n ... (r~)

3.5 sup limsup = T < +00, O<e<1 r-+oo r - r~

where n ... (r) denotes the number of the elements of the sequence {Tk} with

,-,.1 < rj (III , there is a K > 0 such that for sufficiently large x and each t ~ 1, the

number of Tk with x < ITkl :5 x + t is less than Kt; ill " for any fixed 5 > 0, the inequality IITzi -ITk II > e- I"'kI6 holds for sufficiently

large k and any l f kj IV) there is an 0: with 0 < 0: < 71'/2 such that

3.6 largh)1 < 0:.

Assume that the domain n satisfies conditions 0(1) and O(II) given in Section 2. Moreover assume that one of the angle domains defined in O(II), say .6.1, is given by

3.7 71' larg(z) - 71'1 < -:-

2"(

where the constant (al =h > ~. Note that this implies that 0 does not contain the origin O. Moreover, we assume that for any z E 0,

3.8 Izl ~ rn > 0, Tn < ro.

For {An} (n = 1,2, .•• ). assume that it is a sequence of complex numbers with

3.9) IAnl ~ r), > 0, larg(An)1 < a)"

where 0 < 0:), < 00. Since a), can be greater than 71', we think of An as being located on the Riemann surface of the logarithm.

IT the sequence of exponents {Tk}, (k = 1,2, ... ) consists only of strictly positive thus real) terms tending to 00, that is, if F is represented by an ordinary Dirichlet


y

G-!J--1

x

FIGURE 2

series, the classical (R)-order p. and (R)-type u· of F (the Ritt order and Ritt type of F) are defined respectively by (see [171. [221 or [Ill)

(3.10) • li log log M;'(u) p = msup

u-+-oo -u

and when 0 < p. < 00,

(3.11) • lim log MF(u) u = sup O.

u-+-oo e-UP

where MF(u) = sup IF(u + iv)\.

Ivl<oo

It is known (see, for example, [221 and [11, Chapter 21) tha.t, in this case, if

then

(3.12)

and when 0 < p. < 00 and

then

(3.13)

li log k

msup-- < 00, 10-+00 7'10

lim logk = 0, /0-+00 7);

u· = lim sup ( 7);. Id/o\po/Tl:) . 10-+00 ep

We would like to derive similar relations when the entire function F is represented by a generalized Dirichlet series (3.2). But if {7);} contains non-real elementes), the definitions of order and type given above seem no longer suitable. For example, if F(B) = e-n with \7'\ > 0 and arg7' = Q =F 0, we see that for any u < O.

MF(u) = sup e ITI(uC0801-vsiDOI) = +00.

Ivl<oo

For this reason, we modified in [5) the above definitions as follows:

COMPLETENESS OF THE SYSTEM {/(>On")} IN L! OJ 179

Define, for u < 0,

(3.14) MF(U) = sup IF(u + iv)\, I"I$-U

then define the (mR)-order p and (mR)-type (7 of F (the modified Ritt order and modified Ritt type of F) by

(3.15) p = lim sup loglogMF(u) u-+-oo -u

and if 0 0 and argT = 0,0 < 10\ < 11'/2, we have for u < 0,

MF(U) = sup e-1T1(ucosa-1Isina) == e-ITlu(cosa+lsinal) , 11I1$-u

so the (mR)-order of F is O.

Example 4. Let T be a complex nnmber with 11'1 > 0 and argT = 0=11'/4. Consider the entire function

(3.17) 00

-... ~ 1 F(s) = ee -1 = ~. e-S(nT)

n=l n! (s=u+iv).

For the function G(s) = ee-·", we have

IG(s)1 = IG(u + iv)1 = exp [e-1T1(ucos a-"sin a) . cos ( -\TI(usino + v cos 0))]

1 1 = exp exp - y'2IT\(u - v) . cos v'2\T\(U + v) .

So, for u < 0,

MG(u) = sup IG(u + iv)\ :=: IG(u - iu)\ = exp(e-J2u1TI). 1"1$-"

Thus the (mR)-order of G and the (mR)-type of G are p = v'2\r\ and (7 = 1 respectively. From the inequalities \G(s)l-l ::; \F(s)\ ::; \G(s)\ + 1, it immediately follows that the (mR)-order and (mR)-type of F are also p = v'2\rl and (7 = 1.

4. An estimate of Idk \ and a uniqueness theorem

Now we give an estima.te for the upper bonnd of Idk\, where dk (k = 1,2, ... ) are the coefficients of the generalized Dirichlet series (3.2) which represents an entire function F(s).

First we need the following estimate. Its proof will be given in Section 7.

Lemma 4.1. Under the assumptions (1), (II)(3.3) and (3.4), (III)' and (III)", denoting by net) the number of Ti with \Til ::; t, if (a) there is a number p with 0< p < 1, for sufficiently large n,

n D. ( ) <po ; 4.1 n(3\r .. D D*


or (b) for sufficiently large n, n(3\Tn D = n, then we have

(4.2) lim sup log\Tj (,\Ie) \-1 ~ H, Ie-+oo Tie

where

(4.3)

and, for case (a),

(4.4)

for case (b),

(4.5)

with L satisfying LD· > 5K.

Theorem 4.1. Assume that the exponential sequence {Tie} satisfies conditions (I), (II)(3.3) and (3.4), (III)" (III)" and (IV) given in Section 3, and the condition (a) or (b) given in Lemma 4.1, and that p and a are the (mR -order and (mR)-type of F(s), respectively. If ° 0, Jor k sufficiently large,

\d \ < C e(7rD·+H'+2e).Re(TIo) ep a + E: • [ ( )1Re(TIo) P

Ie 1 Re(7),) ,

If p = 0, then, given E: > 0, for k sufficiently large,

[ 1Re TIo E

Idlel < C1e(7rD·+H'+2e).Re(TIo) ~

Re(Tk)

where C1 is a constant

C1 = (7rD· +E:) 100

g(t)e-<7rD'+E)tdt,

with

(4.6) g(r) = g (1 + 1~12). and H' = H / cos 01 with H given by (4.4) or (4.5) corresponding to the condition (a) or (b).

The proof is basically the same as that of Theorem 3 in [51 except for the following two points: (1) We can use D· instead of D. Indeed in [51 it is assumed that the limit limk-+oo k/l7),l= D < 00 exists, but here by [11, Lemma 2.2] and the assumption (II)(3.3), we have

(4.7) log g(r)

lim sup ~ 7r D· . r-+oo r

Hence we can still use [5, Lemma 3.2]) if we use D· instead of D. (2) We need to use the above Lemma 4.1 to estimate l/ITk( -Tk)1 rather than using [5, Lemma 3.1].

We now prove a uniqueness theorem which is a modification of Lemma 1 in [221·

COMPLETENESS OF THE SYSTEM {J(>. ... )} IN L!(O) 1&1

Let {An} be a sequence of complex numbers satisfying (3.9) given in Section 3, l' be a fixed number with 11""(' /2> Cl>., let ~(z) be a. function analytic in a domain

11""(' II) = z: Izl ~ r>., larg(z)l:::; 2

on the Riemann surface of the logarithm, and {An} (n = 1,2, ... ) be its zeroes, i.e., ~(An) = O. Denote by nA(r) the nnmber of the elements of the sequence {An} with IAnl < r. Define, for r > r A,

(4.8) M",(r,'Y') = sup 1~(reI6)1. 191::;"''1' /2

Let B = bcosb be the maximum of the function xcosx in (0,11"/2).

Theorem 4.2. If for some p > 0, U < 00,

lim log Mil>(r, 'Y') 1I"B' 'f b t Cl>. > -j

P

) 1· 'nf n A (r) U 4.11 lID 1 > , r~oo r P 11" cos (ClAP)

'f 11" t Cl>. < 2p'

then ~(z) == 0 for z Ell).

PROOF. Consider the function G(z) = ~(z'Y'), where z'Y' is the branch with z'Y > 0 for :t = X > O. Now G(z) is an analytic function on the domain

1/ ' 11" 11)1 = z: \z\ ~ r A '1 ,\arg(z)\ ~"2 '

and G(A~t") = 0, i.e., bn = Alj'Y' (n = 1,2, ... ) are the zeroes of G(z). Clearly, arg bn )\ < 11"/2. We claim that G(z) = G(rei6) == 0 for z E lIJ)1, 8.nd hence, ~(z) == 0 for z E lIJ). If G(z) ¢. 0 for z (i 11)1, then by Carleman formula (see, for example,

1/ ' {13l , for r>. .., <). < R, we have

4.12) 9n 1

cos ,:::; R 'Y 11"

"./2 log\G(Rei6 ) I cos 9 d9

-". /2

1 R 1 1 + y2 - R2 logIG(iy)G( -iy) I dy + 0(1),

211" A

where 9n = arg(An ), and 0(1) is with respect to R -+ 00 for fixed A. Since for 9 ~ 11"/2, we have that

IG(Rei6 )\ = I~(R'Y' ei'Y'6)\ ~ sup I~(J(Y' ei'Y'6)I 181::;"./2

= sup I~(R'Y' ei<l»I = M/fI(R'Y', ')"), 1<1>1::;".'1' /2

it follows that, by (4.9), given e > 0, and for R sufficiently large, we have

logIG(Rei6)\ ~ (u+e)R'Y'p.


The remaining of the proof is to estimate the upper bound of right hand side and the lower bound of left hand side of the inequality (4.12). By the same arguments as in the proof of Lemma 1 in [22, pp. 105-107], we then obtain

1· ·nf n,\(r) < uo:,\p if b lml -- -- 0:,\ >-; r-toc> rP - 11" B p

and

1· ·nf n,\(r) rP - 1I"cos(0:,\p)

if 11"

0:,\ < 2p'

a contradiction with (4.10) or (4.11). o

5. Completeness of the system {zTJ.}

The next theorem is on the completeness of the system {Z"'·} in L![!lj. We studied this problem in [4], but there we assume a stronger condition for the sequence of exponents {Tk}, namely, the limit

lim ~ = D (0 < D < 00) k-toc> ITk I

exists. Instead of this, in this paper we assume the weaker condition (II given in Section 3.

Theorem 5.1. Assume conditions !l(I), !l(II) for the d rna·n fl, and (I , (II and (IV) for the sequence {Tk} given in Section 3. Jilurthennore assume that

(5.1)

Denote

(5.2)

where

D. - (2+T)(1-coso:) - (1- 2~) > O.

h = max g(x,y), (x ,11) ED

g(x,yl~ J 2x [D.-(2+ T +Yl(l-OOIal -2x-(l--.!..)), (2 + 'T + y)2 sin2

0: + x2l 21"

and

1D> = { (x, y) : x > 0, y > 0, D. - (2 + 'T + y)(l- cos 0:) - 2x - (1- 2~) > o}Let

(5.3)

with eo some positive number. If

(5.4) fOO dr ---=+00 r 1+'1-8 •

then the system {z"'·} (k = 1,2, ... ) is complete in L~ [!l].

,

OOMPLETENESS OF THE SYSTEM (f(~n.z)} IN L!IO}

PROOF. We only need to prove that if f E L![S11 and

(5.5) (I(z), z"'~) = 0, k = 1,2, ... t

then J(z) == O. So we assume that (5.5) holds. By Lemma 2 in [211, condition (11)(3.5) implies that for any h with 0 < h < 2;"" we can find 8 number q > 0 and a sequence {"k} of positive numbers with Vk+1 - Vk ~ q such that the sequence {ILk} = {7'k} U {"k} satisfies

By (IV), clearly

lim k =!.. k-l-oo \ILk I h

And we have (see [21])

(5.6)

Let

and

Denote

00

T(z) = IT k=1

00 e-iB'II 1 l(s) = --211' T(' ) dy,

• 8 = U +lV.

Q'=

Q.., =

-00 ly

. 1 8 = U + ~v : Ivi < 11' - cos a -

h

,

1 1--

21 and for sufficiently small 0 > 0 and 81 > 0, denote

and

. 1 86= s=u+lV:lvl~1I' 'hcosa-8 ,

Q~= 8 = U + ill : Ivi ~ 11' 1 -cosGr-8-h

1 1--

21

Q = 8 = U + iv: \v\ < 11' D .. - (2 + 7' + 81)(1- cos a) - 28-

Let us take h = 1 .

2+7'+81

,

1 1--

21

Then, by (5.1) and (5.6), we will have for 8 and 81 sufficiently small,

1 -oosa-8-h

1- 1 21

- (D~ + 8)

= !. cos a - 28 - 1 _ 1 -!. - D h ~ h *

•

1 = D .. - (2 + 7' + 81)(1- cos a) - 28 - (1- ) > 0 21

184 A. BOIVIN AND C ZHU

Thus, the strip Q is located inside the strip Q~, and the distance from the boundary of Q to the boundary of Q~ is greater than 1I'(D: + 15).

Let z = e~, e = el + ie2 and nf be the image of n in the e plane. By condition n(Il), n f must be located inside Qf. If sEQ..., and ~ E n', s - ~ must be inside S6, so the function l(s -~) is analytic (see (4, Lemma 2.21). If s E Q~ and ~ E 0', l(s - e) must be uniformly bounded. Fix fez) E L~[n], and define, for 8 E ~, the function

(5.7)

As mentioned above, G(8) is analytic in Q..., (hence in Q~) and unif rmly b unded in Q~.

By [4, Lemma 2.4], if for s E Q~ the above G(8) ~ 0, then

(5.8) J l f(z)zn dz = 0, n = 1,2, ....

By Dzhrbasian's theorem, if (5.4) holds, the system {z } n = ,1,2,... is complete in L~[n]. Thus, by the Hahn-Banach theorem, from 5.8 it f 11 Vol'S th t fez) = ° for ZEn. So, we only need to prove that G 8 = 0 for 8 E ct, whenever f satisfies (5.5). We will use per) to denote olrs below.

We will now make use of the sequence {Ilk}. Let

l(z)=II 1-\ =2:";z", 00 ( 2) 00 l

1.=1 Ilk n=O n.

and ,),(z) be the Borel transform of l(z), that is

00 In ,),(z) = 2: zn+l .

n=O

We know that l (z) is an entire function of exponential type 11' D" , and ')' z IS analytic outside the vertical line segment with centre at the origin and length 271" D". For sufficiently small 8 > 0, define the convolution operator

L[y(s)] = ~ r ')'(e s)· y(e) de, 211'l JI~ 81 1f(D·H)

where the function yes) is analytic in Q~. Since the series representing ')'(e - s) is convergent uniformly on e - s =

1I'(D: + 8), we can interchange the order of the integration and the summation as follows:

(5.9) 1 1 (00 In ) L[y(s)] = 2 . 2: (e _ s)n+l . y(e) de 1I'l IE sl 1f(D:H) • 0

-f: ~ r y(e) de = f: In . y(n)(s) n 0211'i lIE 81 1f(D:H) (e - S)n+l n-O n! .

Note that G(s) is analytic and uniformly bounded in Q~, the strip Q is located inside the strip Q~ and the distance from the boundary of Q to the boundary of Q~ is greater than 1I'(D: + 8). So, for SEQ, we can define the function

~(s) = L[G(s)].

COMPLETENESS OF THE SYSTEM {/(~" .. )} IN L!IOl 186

By (5.7), for B E Q~, we have

C(s) =: e 1'.(. e)

e) - L: T' (1L1e) , del d{2 11'.I<t"

+ ~~. e-I'~(·-()

0' II'~ 1 <t" IJ.1e

= Gl.t"(s) + G2•t" (s),

where the sequence {tn } satisfies n 2: t", 2: (1 - >')n while>. is a sufficiently small positive number, Gl •t"(s) and G2 •t"(s) denote the above first and second integra.tion,

Note that, using condition (5.5), we have

J( ee)\ee \2 el'k( del de2 e-I'''·

I#'kl<t" 0'

1 J(z )Zl'k dx dy e-l'kB

o

J(Z)Z"k dx dye-ilkS. o

We claim that for n E N, and for 8 E Q,

5.10) L[G2•t" (s)] = O.

Since C2.t.. (8) is a linear combination of e-lIkS , it is enough to show that L[e-II,,8] = O. Indeed, by (5.9), we have for any kEN,

00

L[e-lIk8] = [L 4 (-IIIe)i] . e-lIkS = l( -Ilk) . e-lIkS = O. . 0 1 . • =

As in [4, pp. 13-14], we have that for Re(s) = u > 0, and SEQ;,

5.lI} (ftoo r2ne-p(r) dr)1/2

\G (s) \ < inf Cn ~o ----;-;c~,____:_....,__!-;---l.t" - "EN \es\(l-A)nsin('J1'I')'

C is a constant independent of s and n, and IL is a small positive Dllmber

6 tan(rrIJ.) < (l/h) sin a •

For 8 E Q, noting (5.10), we have

~(s)1 = IL[G(s)]l = \L[Gl •t" (s)]l

$ Cl (6) . max{IG1 •tn WI: e E Q~, \Re(e) - Re(s)1 $ 7l'(D: + o)},

where Cl (6) is a constant only depending on 6. By (5.11), for SEQ, Re(s) ~ 0 we have

(ftoo r2"e-p(r) dr)1/2 (rOO r 2ne-p(r) dr)1/2 ~(8)1 < inf ~ 0 < inf Cn ~J:.>!;IO--::;;:,.......,.,=--,--!;--

- "'EN 2 le8 /e'J1'(D~+c5)1(1-A)n8in('J1'I') - nEN 3 le81(1-A)nSin(1I'I') ,

where C2 and C3 are constants independent of 8 and n.


Let

and r"

H(r) = sup .jM ' nEN Zn

where f = c\e8 \(1->.) sin(".,..)

with c a constant independent of s and n. By [4, Lemma 2.61, we have for sEQ and Re(s) > 0 sufficiently large,

\<1>(s)\ ~ H~r) ~ exp[-q. p(c\eB\(1->.) Bin ,..,..»)],

where q > 0 is a constant. Now we transform Q (with respect to s) into the upper half-plane Im W ~ 0:

(i) by Wi = e8, Q is transformed into an angle domain arg( WI ~ 11"1 with

(5.12) l = D. - (2 + T + <h)(I- coso) - 20 - (1- 2~} (ii) by Wz = w~/(2I), the above angle domain is then transformed into the right half-plane Re(wz) ~ 0; (iii) by w = iwz, the right half-plane is transformed into the upper half-plane Im(w) ~ O. The remaining of the proof is the same as that in [4, pp. 15 -17] except that the quantity l is now given by 5 12 rather than l = Dcoso - <5 -1 + 1/(2'Y) given in [4, (23)], and correspondly the quantity h is also given by the expression (5.2) rather than that expressed ill [4, 18]. Thus, by the assumption (5.4), we have <1>(s) == 0 for SEQ. Hence G s == 0 for sEQ, and G(s) == 0 for SEQ; since G(s) is analytic in Q; and Q c Q;. The proof is complete. 0

6. Completeness of the system {f(>.nzH

We now present the main result of this paper:

Theorem 6.1. Assume that:

(i) the functions f(z) and F(s) are given by (3.1) and (3.2), respectively, their complex coefficients die =/: 0 (k = 1,2, ... ), and the sequence {TIc} of complex exp0-

nents satisfies conditions (I), (II), (III)" (III)", (IV) and the cond~hon (a) or (b) given in Lemma 4.1;

(ii) the unbounded domain 0 satisfies conditions 0(1) and O(II); (iii) the complex sequence p.n} satisfies condition (3.9).

Moreover assume that

(iv) the entire function F(s) has (mR)-order p and (mR)-type u , with P .(r) > Q>. (2-)PP (pU)8tJ, r~oo rSPP 11" B sa'

or

(6.2) lim inf _>.- > - pu stJ n (r) 1 ( 2 )PP r~oo rSPP 1I"Sp(3c08(Q>.Sp(3) so' () I

:1 b 'J Q>. > -;qi

sP/J

COMPLETENESS OF THE SYSTEM {!(>O",,)} IN L!\nJ 187

where B = b cos b is the maximum 01 the function z COB z in (0, 1r /2) as m 'Theorem 4.2, and (3 = 1/(s - p) > O.

If

and

D. - (2 + 7')(1- cos a) -1

1- -=-2')'

00 dr -:--- = +00, rl+Tl S

> 0,

where '1 is given as in Theorem 5.1, then the system {J(AnZ)} is complete in L~lnl.

PROOF. Consider the function I(wz) with zEn and won the Riemann surface of the logarithm. Clearly, for fixed z, J(wz) is an analytic function with respect to w on the Riemann surface of log w. Now we restrict w to the domain

1r')" JI)) = z: \z\ ~ r", \arg(z)\ ~ 2 '

where "/ is a fixed number satisfying 1r')" /2 > a". It is clear that An E [J) for n = 1, 2, .. ,. ....

Since F(s) = J(e- S) has (mR)-order p and (mR)-type (1, for a.ny A' > (1, and

for u sufficiently large with '1.£ < 0, say '1.£ < -'1.£1 with

1r')" '1.£1 > an + 2 '

we have 10gMF(u) ~ A'e-uP

•

Hence, noting that z = rei9 = e-S = e-(u+iv), for lz\ = r sufficiently large, say r > rh a.nd

we have

1r,,/ \0\ ~ an + 2 '

\J(z)1 = IJ(rei9 )1 ~ eA'rP•

Thus, for \wz\ ~ rl and larg(wz)1 ~ an + 1r,),' /2, IJ( wz)1 < eA'lwiPW.

For fixed wE JI)), letting fh = n n {z : \wz\ < rl} and n 2 = n n {z: Iwz\ ~ rl}, we have (noting that z = x + iy, and rn < ro by (3.8))

6.3)

< -

e2A'IwlPIziP dx dy ~ n2 ro

21rre2A'lwlPrP dr + 00

ro ro 00

e2A'IwlPIziP dx dy n

e2A'IwlPrP (1(r) dr

~ c2ecslwiP + e2A'IwlPrP e-a'r' dr, o

where C2 > 0 and C3 > 0 are constants independent of w. It is clear that, for w E [J)

since wI > r", for \wz\ < rl, and zEn (so \z\ > rn), we have r" < \w\ < rdrn And since larg(wz)1 < an + 1r')" /2, we must have

6.4) \J(wz)12 dxdy ~ Cll n 1


where Cl is a constant independent of w. Thus, by (6.4) and (6.3), we have

fl 1f(wz)12 dxdy ~ Cl + C2e"3Iwl" + loo e2A' 10 .. " e-a'''- dr.

Hence, for any p. with 0 < p. < 0.', we have

J'r If(wzWdxdy ~ Cl + c2e"alwlP + C4 sup· exp[-p.,.a +2A' w Prp}, in .. ~o where C4 is a constant independent of w. As in [3, p. 282], we have

(6.5) fllf(wz)12dXdY < Cl+C2e"alwl" +C4expl2C:Y'" s~{3' A'p s". 8 li1. where.B = 1/(s-p). Hence for any fixed wED, f(wz) E L![nj, and f,\"z E L2[n since An ED for n = 1,2, .... To prove the theorem we only need to prove th t f r any h(z) E L![n], if

(6.6) (I(AnZ), h(z)) = f l f(Anz)h(z) dx dy = 0, n = 1,2, ....

then h(z) = 0 for zEn. So we assume that (6.6) holds. Consider the functi n

(w) = (I(wz), h(z») = f l f(wz)h(z) dxdy, ED,

where h(z) satisfies (6.6). By (6.6), we see that (An) = 0 n = 1,2, ... We need to prove that (w) = 0 for wED. By (6.5), as in [3, p. 283, we h ve, f rED,

(6.7) 1(w)1 ~ Cs [C6 + e"TlwlP

+ Cs exp l c: Y" . S~{3 A p s{J w s "1l where C5, C6, C7, C8 are positive constants independent of w. Thus, by Appendix A in [3), (w) is analytic in D. By (6.7), letting IJ -+ 0.' and A -+ (7, we have

1. log M~(\wkt') < ( 2 )pfJ 1 ( )8 1m sup - .- pq Iwl~oo \w\sP,9 sa' sp{3 •

Thus, by Theorem 4.2 and either condition (6.1) or condition (6.2), we must have (w) = 0 for wED. Then we have for wED,

(6.8) (w) = f l [~d"(WZ)Tt 1 h(z)dxdy = Ed" [f l zTth(z) dxdy1WTlo = 0,

and, since d" '# 0 (k = 1,2,. ,. ), we get

fl zTth(z)dxdy = 0, k= 1,2, ....

Note that Theorem 4.1 is used in the justification for interchanging the integration and summation in (6.8) (see Appendix B in [3]), except that here we need to use the condition p < 8 cos 0.. The remaining of the proof is the same as in [3): by Theorem 5.1, using the completeness of the system {zTt} (k = 1,2, ... ) in L![n], we get h(z) = O. The proof is complete. 0

COMPLETENESS OF THE SYSTEM {/(~",,)} IN L!(OI

7. Proof of Lemma 4.1

First we need a few more lemmas:

189

Lemma 7.1. Under conditions (I) and (11)(3.3), Tn(z) i8 an entire junctiOfl 0/ exponential type 11' D- .

PROOF. By (11)(3.3), given e > 0, there exists an I > ° such that for all i > I, 11'.1> i/(D* + e). Thus, for any R > 0, if Izl < R, we have

Z2 R2(D- +ei -""2 < '2 • 7', '

Hence the infinite product in (4.3) converges uniformly in any bounded domain 01 the complex plane, and Tn(z) is an entire function. For r > 0, let (see (4.6))

Since for Izi = r,

and by (4.7),

hence

where

00

g(r) = II i=1

ITn(z)1 ~ II i=1 i;en

~ g(r),

lim sup log g(r) ::; 11' D* , r-+oo r

li log I MT" (r)1 < D* msup _ 11' I

r-+oo r

MT,,(r) = sup ITn(z)l. Izl=r

The following two estimates can be found in [13, pp. 76-78]:

o

Lemma 7.2. Let ZI," ., Zn be any n complex numbers. Given H with 0 < H < 1. II

n

P(z) = II (z - Zk), k=1

then the inequality

IP(z)1 ~ H n e

holds outside exceptional disks with the sum 01 diameters not exceeding 10H.

Lemma 7.3. II an analytic function I(z) has no zeros in a disk {z : Izl ::; R} and il /(0)1 = 1, then as Izl = r < R,

2r logl/(z)1 ~ - R _ r logM/(R),

where

M/(R) = maxl/(z)l. Izl=R


Let Wi = l7'il (i = 1,2, ... ). For fixed n, denote

P2 = 1--. IT \ Wn \

110,-10 .. 1<1 Wi i;en

Using Lemma 7.2, we can prove

Lemma 7.4. Under condition8 (I), (II) (3.3), (III)' and (ill)", we have, for n 8ufficiently large

(7.1)

where we choose

(7.2) 0= Ln· -5K 2K '

with L satisfying Ln· > 5K.


II Z -Wi IT 1 P(z) = ~ = (Z-Wi) IT

110,-10,,1<1 • 110.-10 .. 1<1 10.-10" <1 W. \;e1\ i;en .;en

Suppose the numerator is a polynomial of degree q. By (III)', we know that q ~ 2K. By (III)", for n sufficiently large and p =F n,

(7.3) IWn - wp \ > e-w ,,6.

Taking H = (1/10)e- 1O,,6, by Lemma 7.2, the inequality

(7.4) I II (z - Wi)1 ~ (1~e e-Wn6) q

110,-10,,1<1 \;en

holds outside exceptional disks with the sum of diameters not exceeding e-1O

,.6.

It is not hard to see that in every exceptional disk there is at least one w, with \wn - Wi\ < 1 (if some disk does not contain such a w" then this disk should not be an exceptional disk since in this disk the inequality (7.4) holds for n sufficiently large). Thus, by (7.3), we know that Wn must be outside these exceptional disks· Hence

I II (wn - W,)I ~ (l~e e-1O,,6) q.

110.-10 .. 1<1 i¢n

When n is sufficiently large, we have (1j'(10e))e-1On6 < 1, hence, noting that q ~ 2K, for n sufficiently large,

(7.5) I II (wn-w')I~ (l~ee-1O"6)2K 110,-10 .. 1<1

,;en

Obviously, we have

(7.6) 1 1 ( 1 ) q (1) 2K -> --> - > - . Itu'!!I<l Wi - 11O,QI<1 Wn + 1 2wn - 2wn

i;en i;en

COMPLETENESS OF THE SYSTEM {f(.~ .... » IN L!IOI

Thus, by (7.5) and (7.6), for n sufficiently large, we have

2K

(7.7) , 5 7

20e W" > e-2K(6+1)w ...

For t > 0, use n(t) to denote the number of 1't with \7'i\ ::; t. For fixed n, let

and

Using Lemma. 7.3, we can prove

W 2

1- " w~ . ,

o

Lemma 7.5. Under conditions (I) and (II)(3.3), given e' > 0, we have, for n sufficiently large,


Q(z) = II

By the proof of Lemma 7.1, we see that Q(z) is an entire function of exponential type ?rD·. Clearly Q(O) = I, and Q(z) has no zeros in \z\ « 3wn (since when 10, > WI we have Wi ~ WI+l, but WI+l > 3wn). Thus, by Lemma 7.3,

When n is sufficiently large (since Q(z) is of exponential type ?rD·),

hence, noting that P4 = \Q(wn )\. we get (7.8). o

We now prove Lemma 4.1:

PROOF. AB before, denote Wi = l7'i\ (i = 1,2, ... ), n(t) the number of 7', with 7', ::; t. For any fixed n, let l = n(3wn ) , denote WLn the nearest left Wi to Wn with


IWi - wnl ~ 1, and WRn the nearest right Wi to Wn with IWi - wnl ~ 1. We have

(7.9) ITn(Tn) I

~ \~(1- ~)\= \,JIj'- ~D\I}l,(l- ~DI ii'n W;::;W(

2

1- ,;

= P1 . P2 . P3 . P4 ,

where Pi, P2, P3, P4 denote the above four products in order. We have estimated P2 and P4 in Lemmas 7.4 and 7.5, so we only need to

estimate Pl and P3· For Pl:

logPl = -log(W1W2'" WLn) + log(wn - WLn)(wn - WL,,-l .•. Wn - Wl 1 =: Pl,l + Pl ,2'

Pl,l = - Ln logwLn + (Ln -1)~ogwLn -logwL,,-11 + (Ln - 2)(logwLn-1 -logwL,,-2] + ... + ~ogw2 -logw11

Ln-l = -Ln log WLn + L j(logwJ+l-logwJ ).

;=1

Since when Wj ~ t < wj+1. net) = j. Thus, we have

Ln-1 l wJ+1 1 Pl,l = -Ln logwLn + 1: j t dt

j=l W,

Ln- l l w'+1 net) l W

l.. .. net) = -Ln log WLn + 1: -dt = -LnlogwL" + --dt.

'lW t Wl t J- ,

Pl,2 = Ln log(wn - Wl) - (Ln -IWog(wn - W1) -log(wn - wa)] - (Ln - 2)~og(Wn - W2) -log(wn - UlJ)l-'"

- [log(wn - WL,,-l) -log(wn - WLn)] Ln 1

= Ln log(wn - wd - 1: j~og(Wn - WL,,_j) -log(wn - WL,,-i+d] ,-1

Ln 1 l wn - w l..,,-.1 .

= Lnlog(wn - wd - E ~ dt. ;_1 Wn-Wl.." .1+1 t

COMPLETENESS OF THE SYSTEM {/(~"z)} IN L![O 193

Let nl(t) be the number of w, with Iw, w~1 $ t and W1 $ Wi < 1/11. ... Since when Wn - WLn J+l:5 t < Wn - wL,,-i' nl(t) - i. we have

L.. I W"-WL,, J nl (t) P1,2 = Lnlog(wn WI) - E dt

;=1 W,,-WL .. j+1 t

"' .. -W1 nl(t) = Ln log(wn - WI) - t dt.

Wn-WLn

Now we have

log PI = Pl ,l + Pl ,2

= -Ln log WL" + WL" n(t)

t dt+Lnlog(wn-Wl)-W,,-W1 n (t)

1 dt

WI =Lnlog 1-~

Wn

W1

-

wn-wu" t Wn- W 1 n (t)

1 dt w .. -wun t

W,,-W1 n (t) 1 dt.

Wn.-WLn t

WI Wn-tul n(wn) - n(wn - t) dt log PI > Ln log 1- - t Wn Wn-WLn

= Lnlog 1-WI WUn n(Wn) - n(x) dx. -Wn Wn -a;

W1

Given g' > 0, for n sufficiently large,

1 1 - WI , og > -c . Wn

By III)'. n(wn ) - n(x) < K(wn - x).

So, we have

log Pl > -c'Ln - K(WLn - WI) > -c'Ln - Kwn.

By the definition of W L". Ln < n, hence

-e'Ln > -e' n.

By (II)(3.3). for n sufficiently large, n < (DO< + £')wn • Hence we have, for n sufficiently large,

7.10) logPI > -e'(D" +e')wn - Kwn :::: -[e'(D* +e') +K]wn.

For P3: First, consider the when the condition (a) holds. Assume that WRn'

wa..+l ••.• ,WI are all the Wi satisfying wR.,. $ Wi $ WE, then

log P3 = -log(wR., wR",+l ... WI) + log[(wR,. - W,,)(WR .. +l - w,,) ... (WI - w,,)]

=: P3,I + P3,2'


Denote n2(t) the number of Wi with Wi $ t and WR.. $ w. $ WI. Denote n3(t) the number of such Wi with Iw. - wn\ $ t. Suppose the total number of WR.,.,WR,,+l1'" ,W, is mn. It is not hard to see that

and

mn = n(3wn ) - R,.. + 1 = l - R,.. + 1-

Similar to that in log P1, we can get

[

I n2(t) P3,1 = -mn log WI + -- dt.

WRn t

m n -l

P3,2 = mn log (WI - wn ) - L j[log(wRn+1 - wn ) -log WR..+1-1 - Wn 1 ;=1

mn-l1wRn+j-wn j =mnlog(WI-Wn)- L -dt.

j=1 WRn+j-l-Wn t

Since when WRn +j-l - Wn $ t < WRn+3 - W n , n3(t) = 3 we have

m n -l [Rn+ -Wn t P3,2 = mnlog(WI-Wn) - L -T- dt

3=1 WRn+ -1-

= mnlog(wl - W n ) - _3_ dt. lWI -

1II n t

WRn -111" t

By (III)" n3(t) < Kt. Thus

log P3 = P3 1 + P3 2 > mn log (1 - wn) - K w - D • " WI ......

Now we estimate the value of :!!!.n. for n suffiC1ently large. B 4.1 in the WI

condition (a), there is a sufficiently small positive number e Wlth e < D. such that for n sufficiently large,

n D* - eo -<p.-=:--..;:. n(3wn ) D* + eo'

Thus, by (3.3) and (3.4), for n sufficiently large,

Hence

Wn Wn I n D* + eo n -=-.-._< . mnlog(l- p) - K(wl - WR.J.

Noting that WR" > Wn hence WI - WR" < 2w", and R,.. > 1, we have, for n sufficiently large

(7.11) logP3 > mnlog(l- p) - 2Kwn = log(l- p)[n(3wn) - R,.. + 1] - 2Kwn

> log(l- p)n(3wn) - 2Kwn ~ log(l- p)(D* + e')3wn - 2Kwn

= -(c'D* + c'e' + 2K)wn,

where d = -3log(1- p). Combining (7.7), (7.8), (7.10) and (7.11), we have for n sufficiently large,

109(P1P2P3P4) > -[K +e'(D* +e') +2K(o + 1) +c'D* +c'e' +2K + 311'D* +e'}wn·

COMPLETENESS OF THE SYSTEM {/(~ .. s)} IN L!(ClJ

Given E > 0, take £' > 0 such that E'(D· + E') + dE' + e' < E. Let

H = 3K + 2K(cS + 1) - 3Iog(1- p)D* + 311'D.,

i.e., by (7.2), H = (L + 311' - 31og(1 - p)/bigr)D*,

then we have for n sufficiently large

ITn(Tn)1 > e-(H+e)Wn,

hence (4.2) holds.

1115

For the case when the condition (b) holds, since l = n, WI = Wn for n sufficiently large, and since, by its definition, wR.,. ~ Wn + 1, we have wR,. ~ W, + 1 > WI. In this case, it is impossible to have a Wi satisfying WRn :::; Wi :::; WI so, for n sufficiently large, the factor Ps should not appear in the product Pl . P2 • Pa . P4 in (7.9), or we should set it to be Ps = 1. Thus, by the above calculation, it is not hard to see that, in this case, the H should be changed to H = (L + 311')D* - 2K. The proof is complete. 0

Acknowledgement. We thank the referee for careful reading of the paper.

References

1. R. P. BC)IIS Jr., FUndamental sets oj entire junctions, Ann. of Math. (2) 47 (1946), 21-32. 2. , Ennre juncnons, Academic Press, New York, 1954. 3. A. Boivin BDd C. Zhu, On the L2 -completeness oj some systems oj analytic junctions, Complex

Variables Theory AppL 45 (2001), no. 3, 273-295. 4. , On the completeness oj the system {ZT .. } in L2, J. Approx. Theory 118 (2002), no. 1,

1-19. 5 • The YIOwth oj an entire junction and its Dirichlet coefficients and exponents, Com-

plex Var. Theory Appl. 48 (2003), no. 5, 397-415. 6. M. M. Dzhrbashyan, Metric theorems on completeness and on the representation oj analytic

junctwns, Uspebi Matem. Na.uk (N.S.) 5 (1950), no. 3(37), 194-198 (Russian). T D. Gaier, Lectures on complex approrimation, Birkhsnser, Boston, MA, 1987. 8 G. M. Goluzin, Geometric theory oj junctions oj a complex variable, Transl. Math. Monogr.,

voL 26, Amer. Math. Soc., Providence, Rl, 1969. 9. L L Ibragimov, On the completeness oj systems oj analytic junctions {F(aloz)}, Izvestiya

AkNi. Nauk SSSR. Ser. Mat. 13 (1949),45-54 (Russian). 10. A. F. Leont'ev, On the completeness oj a system oj analytic junctions, Mat. Sbornik N.S.

31(73) (1952),381-414 (Russian). 11. , Entire juncnons: Series oj exponentials, "Nauka", Moscow, 1983 (Russian). 12. B. Va. Levin, DlStnbution oj zeros oj entire junctions, Revised edition, Transl. Math. Monogr.,

voL 5, Amer. Math. Soc., Providence, Rl, 1980. 13. , Lectures on ent,re junctions, Transl. Math. Monogr., vol. 150, Amer. Math. Soc.,

PlOvidence, Rl, 1996. 14. L F. Lobin, On completeness oj a 81/stelll oj junctions oj the JOI'II' {F(>'nz)}, Doklady Akad,

Nallk SSSR (N.S.) 81 (1951),141-144 (RUSSian). 15. , On completeness oj the system oj junctions {J(~nz)}, Mat. Sb. N.S. 35(77) (1954),

215-222 (Russian). 16. A. L Markushevich, On the basis in the space oj analytic junctions, Mat. Sb. N.S. 17(59)

(1945),211-252 (Russian). 17. S. Mandelbrojt, Selies adherentes, reyularisation des suites, applications, Gauthier-Villars:

Paris, 1952. 18. , Dlnchlet series: Principles and methods, D. Reidel, Dordrecht, 1972. 19. S. N. Mergeljan, On the completeness oj systems oj analytic junctions, Amer. Math. Soc,

TransL (2) 19 (1962), 109-166. 20. W. Rudin, and complez analysis, 3rd ed., McGraw-Hill, New York, 1987.


21. X.-c. Shen, On the closure of {z'7' .. logj z} on unbounded curves in the complez plane, Acta Math. Sinka 13 (1963), 170-192 (Chinese); English trans!., Chinese Math. 4 (1963), 188-210.

22. ___ , Completeness of the sequence of functions U(>..,z)}, Acta Math. Sinica 14 (1964), 103-118 (Chinese); English trans!., Chinese Math. ,. (1964), 112-128.

23. C.-Y. Yu, On generalized Dirichlet series, Acta Math. Sinica 5 (1955), 295-311 (Chinese, with English summary).

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WESTERN ONTARIO, LONDON, ON N6A 5B7, CANADA

E-mail address.A.Boivin:boivin(Quwo.ca E-mail address.C.Zhu:czhu28(Quwo.ca

C.ntft! ele R.echereh •• MILtMmatiquM CRt.! Proc •• ding. and Lecture Note. Volumo Ill, 3010

A Formula for the Logarithmic Derivative and Its Applications

J avad Mashreghi

ABSTRACT. We show how an explicit fOl'mula for the imaginary part of the logarithmic derivative of f, where f is in the Cartwright c\8I!!! of entire functions of exponential type leads to a new integral representation of the Hilbert

• transfollU of loglll and also to a representation for the first moment of 1/12.

1. Introduction

An entire function ! (z) is said to be of exponential type if there are constants A and B such that If(z)l :s BeAI "1 for all z E C. In this note, we are interested in two special subclasses of entire functions of exponential type. Both classes are defined by putting a growth restriction on the modulus of the function on the real line. The Cartwright class Cart consists of entire functions of exponential type satisfying the bOllndedness condition

00 log+l!(x)l d 1

2 X < 00, -00 +x

and the Paley-Wiener class PW contains entire functions of exponential type fulfilling f E L2(JR). The inequality log+I!1 :s ~IJI2 shows that PW is contained in Cart.

In this paper we obtain an explicit formula for c.;:s(J' (t)j J(t)), J E Cart, in terms of nonreal zeros of f and its rate of growth on the imaginary axis. Then we provide two applications of this formula. First, we derive an integral representation of the

• • Hilbert tra.ns£Ol III of logl!l. Secondly, we calculate the first moment of IJI2, where J is the Fourier - Plancherel tra.nsform of !, for functions in the Paley - Wiener space .

•

2. Reminder on representation theorems

In this section we gather Borne well known representation theorems about entire functions of exponential type. These results can be found for example in [1-3].

2000 Mathematics Subject ClasBification. Primary 30D20j Secondary 42A38. Kell words and phraseB. Entire functions, Fourier - Plancherel transform, Hilbert tr8J!sform. This work was supported by NSERC (Canada) and FQRNT (Quebec). This is the final form of the paper.

@2010 American Mathematical Society

197

198 J. MASHREGHI

Let f E Cart and let {Zn} denote the sequence of zeros of f in the UPPer half-plane. Since Ln ~zn/lznI2 < 00 and liDln-+oolZnI = 00, the Blaschke product

Bu(z) = rr(1- Z/~n) 1- z/zn ..

formed with this sequence is a well defined meromorphic function. Let

O'u[fl = lim sup loglf(iy)I , 1/-++00 Y

[fl lim log f(iy)

0'1 = sup . 1/-+-00 Y

In what follows, for simplicity we will write O'u and 0'1 respectively for O'u[/] and O'ilf]. The following theorem is a celebrated result of Cartwright.

Theorem 1 (Cartwright). Let f E Cart. Then, for ~z > 0,

. (i ]00 ( 1 t) ) fez) = ce-10'u" Bu(z) exp - -- + -1 2 log f(t) dt , 11' -00 Z - t + t

where c is a constant of modulus one.

Put r(z) := f(z). Then r E Cart and

[f*] l' log\r(iy)1 lim loglf(-iy) [fl O'u = lIDSUP = sup = 171 •

1/-++00 Y 1/-++00 Y

Moreover, the upper half-plane zeros of r are conjugates of the lower half plane zeros of f, say {wn}n~1! and for the BIMcbke product formed with this sequence we write

BI(z) = rr(I- z/wn). 1- z/w ..

n

Therefore, by Theorem 1,

(1) . (i ]00 ( 1 t) ) r(z)=c'e-10""BI(z)exp - -=-+-1 211ogf(t)dt 11' -00 z t + t

for all ~z > O. We also need the following celebrated theorem of Paley-Wiener. We remind that 1. i. m. stands for the limit in mean and implicitly implies that the sequence is convergent in L2-norm.

Theorem 2 (Paley-Wiener). Let f E PW. Then

10'1

fez) = -0'" j(>..)eiA" d>",

where

. 1 ]N 'At f(>..) = 1. i. m. - f(t)e- I dt N-+oo 211' -N

is the Fourier Plancherel tronsform of f on the real line. Furthermore, the supporting interval of j is precisely [-O'u,O'I).

In particular, if feR) c JR, then l(>..) = j(->..) and thus the supporting interval of 1 is symmetric with respect to the origin, i.e., O'u = 0'1.

LOGARITHMIC DERIVATIVE 199

3. The logarithmic derivative

Let Zn be a point in the upper half-plane. Then the Blaschke factor

b () = 1- z/zn *" Z 1 / - Z Zn

satisfies Ib"" (t)1 = 1 for all t E JR. As a matter of fact, there exists a unique real function argb,,, E COO(JR) such. that

(t E JR),

with a.rg b~n (0) = O. Hence, by taking the logarithmic derivative of bzn , we obtain

d ) b~ (t) 2~zn dt arg b.z., (t = ib:

n (t) = It - znl2'

and thus a.rg b.z., is given by

t 2~Zn t - lRzn lRzn 2) argb .... (t)= I 12ds =2arctan ~ +2arctan ~ .

o S-Zn Zn Zn

Let {Zn} be a sequence of complex numbers in the upper half-plane BUch that

~zn

n

and limn -+oo Zra\ = 00. Let B = TIn bzn . Since the zeros of B do not accnmulate at any finite point of the complex plane, the function B is a meromorphic Blaschke product. In particular, B is analytic at every point of the real line. Hence, for all t E It,

3 B'(t) = 2i ~ ~zn B(t) L.J It - zn\2

n

(t E JR),

and the series is uniformly convergent on compact subsets of lR.

Lemma 3. Let J E Cart. Then, Jor all t E JR,

J(t) - 2 + n \t-(n\2

whel'e {en} is the sequence oj zeros oj J in C \ JR.

PROOF. Let F = J I r. Then, one one hand,

F'(t) f'(t) J'(t) F(t) = J(t) - J(t) = 2i~

J'(t) J(t) .

On the other band, by Theorem 1 and (1),

F(z) = cei(D'I-D'u)z Bu(z) B/(z)

(~Z > 0).

By continuity, this relation holds for ~z ~ O. Hence, we also have

F'(t). B~(t) B1(t) F(t) ;:: 1(0'! - 0'10) + Bu(t) - BI(t) (t E lR).

200 J, MASHREGHI

Thus, by (3),

F'(t) '( ) 2' '" S,)z.. • '" F(t) = 1 0'1 - O'u + 1 LJ It _ % .. 12 - 21 LJ .. .. = i(O'I - O'u) + 2i L It :Z2.12 + 2i L

.. n

= 2{ 0'1 ~ O'u + ~ It :'ZnI2 ).

t-w .. 2

S,)W ..

t-w .. 2

Therefore, comparing with the first formula, we obtain the result. o Note that the real zeros of f do not cause discontinuity in S,) (I' t ! t . Their

effect appears in SR.(!'(t)j f(t)) , An immediate consequence of Lemma 3 and 2 is the following result,

Corollary 4. Let f E Cart. Let {z .. } and {w .. } be respech e y th sequence ! upper and lower half plane zeros of f. Then, for all t E JR.,

fat ~(~~:j) ds = (0'1 ~ O'u)t + ~ ~argbZn(t - ~ ~arg~ t, .. .. where argb is given by (2).

----4. An integral formula for logl!1 and the first moment of i 2

Let {x .. } be a sequence of real numbers such that lim .. --+00 xn = 00 and Xk < x! if k < l. Let {m .. } be a sequence of nonnegative integers. The counting function 1I{"'n} of the sequence {x .. } is defined to be constant between x -1 and x .. and at each point x .. jumps up by mn units. The value of V{:r } t at x 15

not important. For one-sided or finite sequences, lI{z } is defined similarly and it is adjusted such that its value between -00 and the first point of the sequence is zero.

Let f E Cart. In [4], we showed that

~(t) = -11'11{Xn}(t) + (O'u; O'!)t - ! ~argb .... (t) - ! ~argbw t, .. n

where ,..., stands for the Hilbert transform. This formula has been used to obtain a partial characterization of the argument of outer functions on the real line [5]. By a standard technique, one can shift all zeros of ! in the lower half plane to the upper half-plane without changing Ilion the real line. Furthermore, one can multiply I by e-IO'I", to get a new function with the same absolute value on the real --line, but instead 0'1 = O. Therefore, to find 10gl!l, without loss of generality we can assume that f has no zeros in the lower half plane and besides 0'1 = O. Therefore, by Corollary 4, we find the following formula for the Hilbert transform of loglfl.

Theorem 5. Let I E Cart. Suppose that f has no zeros in the lower half-plane and that O'! = O. Let II denote the counting function of the sequence of real zeros of f. Then

~(t) = -7rv(t) -fa' ~(~~:]) ds.

LOGARITHMIC DERIVA'l'IVE 201

Another consequence of Lemma 3 is an explicit formula for the first moment of ljl2, in terms of Uu ! UI and Donreal zeros of f, for functions in the Paley Wiener space PW.

Theorem 6. Let f E PW. Then 171 00

If(t)12 dt, -00

where {(n} is the sequence of nonreal zeros of f. ,

PROOF. Since >.f(>..) E Ll(JR), the Fourier Plancherel transform of f'(t) is i)..j()..). Thus, by the Parseval's identity,

00 171 •

f'(t)f(t) dt = 271" i>../(>..)/(>..) d>" = 271" -00 -a .. -<7 ..

The right-hand side is purely imaginary. Therefore, by Lemma 3, 00

>'I/C>'W d>" = -a .. -00

--

;) f'(t) f(t) dt = 00;) f'(t) If(tW dt f(t) -00

If(t)12 dt. o

Acknowledgements. The author deeply thanks the anonymous referee for his/her valuable remarks and suggestions.

References

L R. P. Boas Jr., Enure funcuons, Academic Press, New York, 1954. 2 P. Koosis, The !oganthmic integral. I, Cambridge Stud. Adv. Math., vol. 12, Cambridge Univ.

Cambridge, 1988. 3 B Ya. Levin., Distnbuuon of zeros of entire functions, Revised edition, Thans\. Math. Monogr.,

voL 5, Amer. Math. Soc., Providence, RI, 1980. 4.. J Mashregbi, Htl!lert transfonn of loglfl, Proc. Amer. Math. SOl:. 130 (2002), no. 3, 683-688. Ii J Masl:.regbi and M. R. Pouryayevali, Argument of outer functions on the real line, minois J.

Math. 51 (2007), no. 2, 499-511.

DEPAKI'EMENT DE MATHEMATIQUES ET DE STATISTIQUE, UNIVERSITE LAVAL, QUEBEC, QC

GIV OA6, CANADA

ErlnaUadd~6: javad.mashreghi~at.u1ava1.ca

o-nut dt Redlerch .. M"th6m .. tlque. CRM Proceedlnl!l a.nd Lecture Now Voluml 51, 2010

Composition Operators on the Minimal Mobius Invariant Space

Hasi Wulan and Chengji Xiong

ABSTRACT. Two sufficient and necessary conditions are given for q, to ensure the composition operator C., to be compact on the minimal Mobius invariant space. Me·mwhile, our results show that some known resuits about the compactness of C., on the Beaov spaces are stiJI valid for the minimal Mobius invariant space.

1. Introduction

Throughout this paper lJ)) will denote the open unit disc in the complex plane C. The set of all conformal automorphisms of lJ)) forms a group, called Mobius group and denoted by Aut(lJ))). It is well-known that each element of Aut(J[])) is a fractional transformation tp of the following form

'(} tp(z) = e' O"a(z), a-z

O"a(z) = 1 -, -az where 8 is real and a E lJ)). Denote by dA the normalized area measure:

1 dA(z) = - dx dy, z = x + iy.

11" Let X be a linear space of analytic functions on J[]) which is complete in a norm

or seminorm 1I·lIx. X is called Mobius invariant if for each function 1 in X and each element tp in Aut (lJ))), the com position function 1 0 cp also lies in X and satisfies that 10 tp Ix = IIll1x; see [2]. For example, the space HOO of bounded analytic functions I on lJ)) with the norm 1111100 = sup{ll(z) I : z E J[])} is Mobius invariant. BMOA, the space of a.nalytic functions f on J[]) for which

1 271"\ i6)12 1 -lal2 I ( )12

sup 211" 0 f(e 11- aei6 12 dO - 1 a : a E J[]) < 00,

is Mobius invariant. Actually, some other spaces of analytic functions on J[]) such as Qp and QK spaces are Mobius invariant, too. See [2-4]. However, the Hardy

2000 Mathem.a.tu:s SubJect Classification. 30D45, 47B38. Key woma and phrases. composition operator, minimal Mobius invariant space, Be80v

spaces The authors are partially supported by NSF of China (No. 10671115), RFDP of China

No 20060560002) and NSF of G1!angdong Province of China (No. 06105648), This is the final form of the paper.

@l2010 Amerlc .... Mathematical Society

203

204 H. WULAN AND C. XIONG

spaces HP are not Mobius invariant. Now we return to our primary interest, the Besov spaces.

For 1 < p < 00 the space Bp consists of all analytic functions / on JI} for which

(1.1)

For p = 00 the requirement is that the quantity

(1.2) sup(1-lzI2 )1!,(z)1 ZED

be finite. When 1 < p < 00 the space Bp is called the Besov space and Boo = B is called the Bloch space. The seminorm II·IIB" on Bp is the pth root of the left of (1.1) if 1 1, we define the Bl by ther way since (1.1) does not converge when p = 1 for any non-constant functi n.

Arazy, Fisher and Peetre [2] defined Bl as a set of those analytic functions / on ID> which have a representation as

00

(1.3) J(z) = ~::>kaak(Z), k=l

00

ak E ID> and L Ck < 00.

k=l

Since a function / could conceivably have several such representation, the norm of BI can be defined by

By [2] we know that the space BI is the minimal Mobius invariant space since it is contained in any Mobius invariant space X. Also, we say that the Bloch space B is the maximal Mobius invariant space; see [7].

We know that for 1 < p < 00 a. function / belongs to Bp if and only if the seminorm

(1.4)

Arazy, Fisher and Peetre showed in [2] that there exist constants C]. and o.a such that

(1.5) cIII/1l ~ 1/(0)1 + 1/,(0)1 + iolf"(z) 1 dA(z) ~ cIII/II·

Hence, (1.4) and (1.5) do permit us to pass the case p = 1 and the connect the space Bl with the Besov spaces Bp. We define now the seminorm of BI as

(1.6) 1I/IIB1 := io1f"(z)ldA(Z) < 00.

Modulo constants, Bl is a. Banach space under the norm defined in (1.6).

COMPOSITION OPERATORS ON THE MINIMAL MOBIUS INVARIANT SPACE 205

2. Composition operators on Bl

Let tP be a holomorphic mapping from D into itself and f e H(lD), the set of ell analytic functions on D. Then tP induces a composition operator Cq,: f -+ f 0 tP on H(lD). Tjani [9] gave the following result.

Theorem A. Let tP be a holomorphic mapping from lD into itself and 1 ensures the composition operator Cq, to be compact for the critical case p = 1. This paper mainly answers this question.

Theorem 1. Let tP be a holomorphic mapping from lD into itself. Then Cq, is compact on Bl if and only if

2.1) lim IIC</'o"0IlB1 = o. \0\-+1

PROOF. NECESSITY. Assume that C is compact on B1. We have that Cq,O"o B1 = I C(O"o - a)IIB1 -+ 0 as lal -+ 1 since O"o(z) - a --+ 0 uniformly on

compacts of D. Thus (2.1) holds.

SUFFICIENCY. We first show that (2.1) implies that

lim sup I(O"aor/>(Z))"ldA(z) =0. r-+1 0ElI) \(z)\>r

2.2

In fact, by (1.5) we see that (2.1) gives that

2.3

Hence

1I)

I(O"a 0 r/>(z))"1 dA(z) $1

f r some a E JI)). It follows that O"a 0 r/> E B1 for some a E lD. Since B1 C B2 = V and (1a is analytic on ii), we have tP = O"a 0 O"a 0 tP E B1 • Note that by (2.3) we know that for given £ > 0 there exists a 0 > 0 such that

sup 1 (O"a t!l r/>(z))"1 dA(z) < e 6<\0\<1 \(zl\>r

f r all r E (0, 1). Letting r -+ 1- gives

$ sup 10"~(r/>(z))IIr/>'(z)12 dA(z) + sup 0$6 (z)l>r \a\$6 \(zl\>r

100~(tP(z)) 11¢>"(z)1 dA(z:

::; C ItP'(z)12 dA(z) + Ir/>"(z)I dA(z) < e, q,(z) >r \(zll>r

where

C = max sup 100~(z)l, sup 100~(z)1 . lal $6 101:56 zElI) zEII)


Note that c!> E Bl and c!> E 'D have been used in the above estimate. Thus we show that (2.2) holds.

Now we prove the compactness of C4>' For any bounded sequence {In} C BIt without loss of generality, we assume that In converges to zero uniformly on any compact subset of JI]l and IIfnllB1 $ 1. To end our proof it suffices to show that Ilfn ° c!>IIB1 -+ 0 as n -+ 00 since Ifn ° c!>(O)I+I(fn 0 c!»'(O)I-+ 0 as n -+ 00. We write

00

fn(z) = LCn,IeUan.~(z), Ie=l

with 00

\lfnllB1 $ LICn,1e1 $ 2, n = 1,2, .... Ie=l

Using (1.5), it suffices to prove

1\ (In(c!>(Z))''\ dA(z) -+ 0, n -+ 00.

By (2.2) for given E > 0, there exists an r, 0 < r < 1 such that for all a ED

sup 1 \(Uaoc!>(z))"\dA(z) $~. aEIIli 14>(z)l>r 2

Hence

1\ (In(c!>(Z))''\ dA(z)

=1 \(In(c!>(Z))''\dA(z) + 1 \ (In(c!>(Z)) , dAz 14>(z)l::;r 14I(z)l>r

$1 \(In(c!>(z))''\dA(z) + 'f:ICn,1e1 1 \(O',,",,(c!>Z))' dAz) 14>(z)l::;r Ie=l 141(2:) >r

$1 \ (In(c!>(Z))''\ dA(z) + E. 14>(z)l::;r

Notice that

1 \(In(c!>(Z))''\dA(z)-+O 14>(z)l::;r

as n -+ 00. We obtain \lIn ° c!>IIBl -+ O. The proof is completed.

Arazy, Fisher and Peetre (2) obtained following theorem.

o

Theorem B. The composition operator C. is bounded on Bl if and only if

and

sup f\O'~(C!>(z))\Ic!>"(z)ldm(z) < 00. aED lID

Now we show a similar result for compact opera.tors.


Theorem 2. Let ¢ be a holomorphic mapping from 11) into itself. Then C</> is compact on Bl if and only if the following two expressions are true:

(2.4) lim 100~(¢(z)) 11¢'(z)12 dA(z) = 0 lal-+l ID

and

(2.5) lim 100~(¢(z))¢/I(z)1 dA(z) = O. lal-+l D

PROOF. Since

it is easy to

C</>O"a B1::::l 10": (¢(z)) (¢'(z))2 + O"~(¢(z))¢/I(z)1 dA(z), ID

that (2.4) and (2.5) imply

lim IIC</>O"aIlB1 = o. a -+1

By Theorem 1, C</> is compact on B1•

Conversely, as')ume that C</> is compact on B1• By Theorem 1 we have

lim C</>O"a B1 = lim I (O"a 0 ¢(z))"1 dA(z) = O. a -+1 la -+1 II)

Since 0" is zero-free and analytic in j[J), we can find a function fa analytic in llJ)

v.ith J 0 = 0 such that (f~)2 = 0":. So fa E B2 and IIfallB2 is bounded. By the estimate

C</>O"aIlB2 :5 CIIC</>O"aIlBl and the assumptiOn, we know that C</> is compact on B2 by Theorem A. A direct computation gives O"~ w) -t 0 in lDl as lal -+ 1. Hence fa tends to 0 uniformly on any compact subset of lDl. Thus IC</>faIlB2 -+ 0 and

lim C~f B2 = lim 100~(¢(z)) 11¢'(z)12 dA(z) = O. a -+1 al-+l ID

On the other band, since

O"a 0 r.p)" = 0": (r.p) (¢')2 + O"~(cp)¢", v.e have

IT,. q, z ¢' z dA(z):5 I(O"a 0 r.p)"(z)1 dA(z) + 10":(¢(z))II¢'(z)12 dA(z). D II) II)

By 24 and the assumption we obtain (2.5). We complete the proof. 0

3. Composition operators between Bl and Bp

Theorem 3. Let r.p be a holomorphic mapping ofllJ) into itself and 1 O"a i1Bp = o. a -+1

PROOF. C~ is compact from B1 to B,p' Choose O"a(z) - a e B1 which converges to 0 unifoIlllly on any compact subset of llJ). Thus

lim C"O"aIIB = lim IICq,(O"a - a)IIBp = O. a -+1 p lal ..... 1


Conversely, for the case 1 IIB" ~ 0 as n ~ 00. Let

with

00

fn(z) = L Cn,kO'a ... ~ (z), k=1

00

II/nllB1 ~ Llen,kl ~ 2, k=1

n= 1,2, ....

Similar to the proof of Theorem lone can prove that (3.1) implies that

(3.2) limsup ( I(O'ao4>(Z»)"IP(1-lzI2)2P-2dA(Z) =0. r~1 aEI!) J1q,(z)l>r

Thus, for given E > 0, there exists an r, 0 < r < 1 such that for all a E D

sup ( I(O'ao4>(z»)"nl-lzI2)2P-2dA(z)~ ~1' aEI!) J1q,(z)l>r

Therefore, by Holder's inequality

{ \ (In (4)(Z)))" \P (1-lzI2)2P-2dA(z) J1q,(z)l>r

= { \fen,k(O'an,k(4>(z»))"IP(I-lzI2)2P-2dA(z J1q,(z)l>r k=1 I

~ (flen,kl)P-1 { f\en'kl\(O'Qn.~(4>(Z»))IT (1- z2 2p-2 dA Z

k=1 J1q,(z)l>r k=1

~ (flcn,kl)P sup { \ (O'a(4>(Z»))'T (1- Z 2)2p-2 dA z ~ E 2. k=1 aEI!) J1q,(z)l>r

On the other hand, we have

{ \(In(4>(Z)))''r(I-\z\2)2P-2dA(z)<E 2 JIq,(z)l~r

as n ~ 00. Hence

lim 1 \ (In (4)(z)) )"\P (1_lz\2)2p-2 dA(z) = O. laH1 D

Thus, Cq, is compact from B1 to Bp. For the case p = 00, if (3.1) holds, then

(3.3) lim IICq,O'a\\8 = O. laH1

By Theorem I, Cq, is compact from B to B. Since B1 C B, Cq, is compact from B1 to B. The proof is completed. 0

Combining our theorems with Tjani's result, we are able to build the following new theorem.

Theorem 4. Let 4> be a holomorphic mapping from lD> into itself and 1 ~ p ~ q ~ 00. Then the following are equivalent:

(a) Cq,: Bp -+ Bq is a compact operator.


References

1 J. M. Anderson, J. Clunie, and Ch. Pommerenire, On Bloch /unction.! and normal /unction.!, J. Reme Angew. Math. 270 (1974), 12 37.

2 J Ara:r.y, S. D. Fisher, and J. Peetre, Mobius mt/anont /unction spoces. J. Reine Angew. Math. 363 (1985), 110 145.

3. R. Aulaskarl., J. Xiao. and R. H. Zhao. On subspaces and subsets 0/ BMOA ond UBC. Analysis 15 1995), no. 2. 101 12l.

4. M. F:s,;;e,l and H. WuJan , On onalytlc ond meromorphlc /unction.! and 8pOce, 0/ QK-type. illinois J. Math. 46 (2002), no. 4, 1233 1258.

So K Madlg&ll A. Matheson, Compact composItion operators on the Bloch space. Trans. Amer. Math. Soc. 847 (1995). no. 7, 2679 2687.

6. J Peetre, New tho gilts on Besot} spaces. Duire Univ. Math. Ser., vol. I. Mathematics Department, Du\re Umversity. Durham, NC, 1976.

T L. A Rubel and R. M. Tunoney, An e:tt.emal property 0/ the Bloch space, Proc. Amer. Math. Soc. 15 19'"'9, no. I, 45 49.

8. J H Shapiro, Componbon opel'lltol"S ond classICal /unction theory, Universitext Tracts Math., 'i>nnger, New Y rk, 1993. t. 1)al11, C mpac compos.hon opemtol"S on Besol} Spaces, Trans. Amer. Math. Soc. 355 2003 , DO. 11 46S3 4698

N Zorboska, C mponhon operators on weighted Dlnclilet spaces, Proc. Amer. Math. Soc. 126 8 no. 7 2013 2023.

D RH ENT F!II THt:r. ATICS, SHANTOU UNIVERSITY, SHANTOU, GUANGDONG 515063, PEO·

s REPl BU OF CHINA

E- '1ddte.ss, H WlIlan wulanOstu.edu.cD E- C Xl ng chengJ i..xiongl!lyahoo. com

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CRM ProeHdlna. and Lec\"re Not81 Volume &1, 2010

Whether Regularity is Local for the Generalized Dirichlet Problem

Paul M. Gauthier

ABSTRACT. We give an example which shows that a regular boundary point for the classical Dirichlet problem need not be regular for the genera.iized Dirichlet problem.

Let G be a bounded open set in lRn. The classical Dirichlet problem for G is the problem of the existence, for every continuous function cp on {)G, of a harmonic function u in G having !p as boundary values. To solve the Dirichlet Problem, Lejeune Dirichlet introduced a variational method, which asserts that a solution u can be attained as a minimizer of the Dirichlet energy in a certain function space. Bernard Riemann na.med this method the Dirichlet Principle and assumed that such a minimizer exists. However Karl Weierstrass, in 1870, provided a counterexample to the existence in general of a minimizer. In 1899, David Hilbert gave a rigourous solution to the Dirichlet problem by justifying the Dirichlet principle, under certain oonditions, thereby foreshadowing the introduction of Hilbert space.

The Dirichlet problem is attacked by somehow providing a candidate u'" for 8. soluti n. Let us call such a candidate a generalized solution. Once we have a generalized solution uCP' there remains the problem of showing that u'" has the desIred boundary values!p. For any continuous function cp on {)G, the Perron method provides a generalized solution, which we denote by u~ and call the Perron solution.

A boundary point p E 8G is said to be a regular point for the (classical) Dirichlet problem for G, if for each continuous function cp on {)G, the Perron solution u~ has the desired boundary behavior at p. That is,

1 lim u~(x) = cp(P). "'-tp

Thus, it is a tautology to say that the classical solution to the Dirichlet problem exists for a bounded open set G if and only if each boundary point is regular.

Of course, if !p is not continuous, then there is no solution to the classical Dirichlet problem for the boundary data cpo However, the Dirichlet problem can be

2000 SubJect C/a.ss'ficatwn. 31B20, 31B05, 31A25, 31A05. Key words and phrases. Dirichlet problem. IWwarch supported by NSERC (Canada). TIllS is the final fO! m of the .

@2010 American Mathematlc"l Socletl

211

212 P. M. GAUTHIER

generalized as follows. For many functions cp defined on aG, but not necessarily continuous, Perron's method still produces a harmonic function u~, which it is natural to call a Perron solution to the Dirichlet problem for the boundary function cp. In some sense, the Perron solution is the harmonic function which makes the best attempt at being a classical solution. If a classical solution exists, then the Perron solution exists and coincides with the classical solution.

Marcel Brelot has shown that the Perron solution u~ exists if and only if .p is integrable with respect to harmonic measure J.L~, for some (equivalently, f revery a E e. Moreover, we have the integral representation

u~(a) = 1 cpdJ.L~, a E G. 8G

Norbert Wiener showed that a boundary point p E ae is regular f r the lassical) Dirichlet problem if and only if the complement of G is not th n at p. F r example, if the complement of G contains a cone with vertex at p, then p is a regular point.

Of course this implies that regularity is a local condition. The regularity r non-regularity of a boundary point p E ae depends only on the nature f the pen set G near the point p.

Our definition of regularity (which is the usual one) is f r the clllSS1cal Dmchlet problem. Now that we have introduced the more general Perr n solut' n to the Dirichlet problem, it is very tempting to think that regularity is I al also ill terms of Perron solutions. Namely, one might think that if p E aG is regular f r the (classical) Dirichlet problem, then (1) holds whenever u~ makes sense and cp is continuous at p.

The purpose of this note is to provide a counterexample. This example ",-as formulated in a conversation with Aurel Cornea over 30 years ago.

Example 1. There exists a bounded simply connected domain D in ]R2, having the point (0, 0) as a regular boundary point for the (classical Dirichlet problem and containing the interval {(x, 0) : a < x < I}, and there exists a functton cp on aD integrable with respect to harmonic measure (so the Perron solution u~ exists and a neighborhood U of (0, 0) in R2 such that: cp(x, y) = 0 for (x, y E un aD, but

limsupu~(x,O) = +00. :1:'><0

PROOF. It is easy to give an example of an open set G having the required properties, except that it is not connected. We shall then add (carefully chosen) connecting channels between components of G to obtain the desired domain D.

For n = 0,1,2, ... , let Rn be the open rectangle

Rn = {(x,y) E]R2: 2-n - 1 < x < 2-n ,lyl < I}

and denote by Hn = {(x,y): 2-n - 1 < x < Tn,y = ±1}

the upper and lower boundary segments of Rn. Let (xn , 0) be the mid-point of the rectangle Rn. Thus, Xn = (2-n 1 + 2 n)/2).

Set G = Urn Rn. We now define the function cp on aGo First of all, we put cp(x,y) = a on all vertical boundary segments {(2-n ,y) : lyl $ I}, n = 0,1,2, ... , and {(O,y) : Iyl $ I}. On horizontal boundary segments {(x,±I) :

WHETHER RIllQULARITY IS LOCAL FOR THE GENERALIZED DIRICHLET PROBLEM 213

2 " 1 < % < 2 n}, n - 0,1,2"", we set 1,0(%, ±1) = An, where An > 0 is chosen so large tha.t the value of u~, a.t the mid-point (%n, 0) of the recta.ngle Rn, is greater than n:

(2)

The open set G and the boundary function 1,0 have all of the required properties with the exception tha.t G is not connected.

We shall now construct a domain D from G. For each n = 1,2,.", let 8n be a segment

8n = {(Tn, y) : Iyl < En},

for some 0 < En < 1 to be chosen later. Set n n

Dn= U Rku U 8k k=O k=l

and 00 00

D = U RkU U 8k. k=O k=l

By a.buse of notation, we denote by u: the Perron solution of the Dirichlet problem on Dn , with boundary values 1,0 restricted to 8Dn . This makes sense, since 8Dn C

aGo Let P~" and P~,,, denote harmonic measure for the domains D and Dn at a point a, b respectively in D or Dn.

From the maximum principle,

JL~,"(HI U 82 ) < p~(81)' (x, y) E Ro. Thus, we may choose £1 so small, that

1 ( 1 3 AI' Pxo,o HI U 82 ) < 21 .

We note that on Db by the maximum principle,

p~II(Hl) < J..L~,"(H1 U 82),

and so, by 3,

.(

Suppose, for j = 1,2, .. , ,n - 1, we have defined Ej such that

D 1 Aj • J..Lxo,0 (H, ) < 2j'

We may choose IOn SO sma,ll, that

5 -. We note that on Dn , by the maximum principle,

J..L~II(Hn) < J..L~,"(Hn U 8n +1),

and so, by (5),

6

214 P. M. GAUTHIER

Thus, by induction, (6) holds for all n = 1,2, ... , and 80

r 'Pdll~.o = tAn' p.~,o(Hn) < -'0 '1l~,o(Ho) + 1 < +00. laD n=O

Hence, 'P is integrable with respect to harmonic measure for D. Therefore, the Perron solution u~, by which we mean the Perron solution for the restriction of 'P to aD, exists.

It follows from Theorem 6.3.6 in [I} that 'U~ > 'U~ on Rn and so by the maximum principle and (2) it follows that

U~(Xn,O) > n, n = 0,1,2, ....

Thus, the Perron solution u~ has all of the required properties. o References

1. D. H. Armitage and S. J. Ga.rdiner, Classical potentJal theoT1/, Spnnger Monogr. Math. Springer, London, 2001.

DEPARTEMENT DE MATHEMATIQUES ET DE STATlSTlQUE, UNIVERSITE DE MONTREAL CP 6128

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