Analytic functions of bounded mean oscillation and the Bloch space

Integr Equat Oper Th Vol. 17 (1993)

0378-620X/93/040501-1551.50+0.20/0 (c) 1993 Birkh~user Verlag, Basel

A N A L Y T I C F U N C T I O N S O F B O U N D E D M E A N

O S C I L L A T I O N A N D T H E B L O C H S P A C E

PRATIBHA G . GI tATAGE AND DECHAO ZHENG

ABSTRACT. In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch space ~ is the image P(Ll) of space of all continuous functions on the maximal ideal space of H ~176 under the Bergman projection P. It is proved tha t the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U) .

I n t r o d u c t i o n .

Let D be the open unit disk in the complex plane and OD the un i t circle. Suppose u is an

integrable funct ion on cOD and I is a subarc of cOD. We denote by u(I) the average of u over I,

1// where [I] is the usual Lebesgue measure of I on OD. The quant i ty

1 f ]/I~ - ~ ( O I d O

represents the mean oscillation of u over I. We say tha t u is a function of bounded mean oscillation

if

I1~11. -~ sup 1 ~ [u - u(I)ldO < oo

the supremurn being taken over all subarcs I C cOD. The set of all funct ions of bounded mean

oscillation will be denoted by BMO. It is clear tha t L~176 dO) is a subset of BMO.

We define BMOA to be the set of functions f6 BMO whose Poisson extensions to D are analytic.

As is well-known, f is in BMOA iff

sup f I f (S(8)) - f(S(O))ldO < oa. SEAut(D) J

It is easy to see tha t BMOA is a confox'mal]y invariant Banach space of analyt ic functions with

norm

[IIIIBMo = f(O) + Ilfll* (see [7], [12, Chapte r 6, Section 1], and [26]).

502 Ghatage and Zheng

Another interesting conformally invariant Banach space of analytic functions is the Bloch space

/312], [20]. This consists of all analytic functions f in D such that

Ilfll/3 = sup(1 - 1~12)lf'(z)l + If(O)l zED

is finite. The little Bloch space/30 is the subspace of/3 which contains functions such that

(1 - I z l 2 ) f ' ( z ) ~ o

as Izl--* 1. Let dA denote the usual normalized area measure on D. The Bergman space L 2 is the subspace

of the Hilbert space L2(D, dA) which contains analytic functions on D. So there is a projection

P, called the Bergman projection, from L2(D, dA) onto L 2. It is well-known that P extends to

LP(D, dA) as a bounded operator for 1 < p < oo [4].

The algebra of bounded analytic functions on D is called H ~176 Le t /2 denote the C*-subalgebra

of L~176 dA) generated by H ~176 Since BMOA is the image of bounded harmonic functions on D

under the Bergman projection a n d / 3 = P(L~176 dA)), we have that

H ~176 C BMOA C P(/2)C/3.

On page 40 of [4], Axler raised the question whether P(/2) equals the Bloch space/3. Recently

Axler and Zhu [6] proved that f is in P(/2) iff (1 - ]zl2)f'(z) is in /2 . So P(/2) is a proper snbspace

of/3.

On the other hand Jones gave the following distance formula. For f in /3 ,

dist/3(f, BMOA) ~ inf{e, (1 - I z l 2 ) - l X a , ( / ) d A is a Carleson measure}

where f ~ ( f ) = {z E D, (1 - Izl)lf'(z)l > ~} and xn~162 denote the characteristic function of the

set ~c( f ) (see page 14 of [2]).

Another motivation to find the closure of BMOA in the Bloch space arises naturally in the lifting

of Hankel operators from the Hardy space to the Bergman space [13].

In this paper we will show that the closure of BMOA in the Bloch space is precisely P(U). On

one hand our result shows that P(U) is a very "small" subspace of the Bloch space. On the other

hand we present another characterization of the closure of BMOA in the Bloch space.

A theorem of Ahlfors-Weil, and elementary distortion theorem show that a function f is in

the Bloch space iff it is representable in the form f(z) = alogg'(z) for some constant a and

some univalent function g that extends quasiconformally fl'om D to all of C. Thus the quasicircle

g(OD) satisfies the diameter condition of Ahlfors ([1] [21]). Pommerenke found the analogous

characterization for BMOA [22]. It turns out that f is in BMOA iff f has the above form with

g(OD) quasi-smooth, that is, g conformMly maps D onto a Jordan domain whose boundary satisfies

a certain arc length condition first studied by Lavrentiv. By results of Makarov we will show that

if f is in P(/2), then the f has the above form and g(D) is not (~) Makarov domain for any ~ > 0.

This leads to more examples of functions which do not belong to P(/2).

The authors would like to thank Sheldon Axler, Christopher Bishop and Peter Jones for many

helpful conversations.

Ghatage and Zheng 503

C a r l e s o n m e a s u r e , B M O A and i n t e r p o l a t i n g s e q u e n c e s .

For z in D, the M5bius transformation r : D ~ D is defined by

Z - - W

r = 1 - ~ w "

A positive measure # on D is calIed a CarIeson measure if there is a constant N such that

~ ( Q ) < N h

for all squares

Q= {re iO :O0 < O < 0 0 + h , l - h < r < 1}.

The conformally invariant character of a Carleson measure is reflected in the following description.

A positive measure # on D is a Carleson measure iff

s~p / ~--~d~(z) < 0r z o e D J D [1 - - ~ o Z I

We define a positive measure # on D to be a vanishing Carleson measure if

Iz01~1 ]1 2 d#(z) = O.

BMOA can be characterized in term of the Carleson measure. Fefferman has shown that for

an integrable and analytic fl~nction f on D, f is in BMOA if and only if (1 - [z]2)[f'(z)]2dA is a

Carleson measure [12, page 240], So (1 - ]z]2)-~Xa,(l)dA is a Carleson measure for any e > 0 i f f

is in BMOA since ( l - H 2 ) - I X f ~ , ( I ) d A < (1 - Iz I2) ] f ' (z ) I2dA

- - ~2

This may be motivation that led to Jones's theorem. We would like to thank Peter Jones for

permit t ing us to present the following proof based on some of his ideas.

J o n e s ' T h e o r e m . Suppose f is in the Bloch space. Then the following quantities are equivalent.

(A) distB(f, BMOA); (B) inf{e : Xfl,(])dA--4L~,_lz,2 is a Carleson measure.) where Ft,(f) = {z : z e D, (1-]zl2)lf'(z)[ _> e},

X denotes the characteristic function of a set.

P r o o f . Let f be in the Bloch space. We may assume that f (0) = i f (0) = 0. Define

and

(1 -[w12)f ' (W)dA(w~ f l ( ~ ) = (1 - z ~ ) 2 ~ ' ' ,(f)

f2(z) = /D\n, ( ] ) (1-- Lw]2)f'(W)dA(w~ ( 1 - z ~ ) 2 ~ , J'

It follows from Lemma 4 in [6] that

f ( z ) = f , ( z ) + k ( z ) .


Let f3(z) = f2(z) - f2(0) - f~(O)z. To show that the Bloch norm of f3 is bounded by 6e, let z in

D. Then

/0 f ; ( . ) = ~ Z~'(tz)dt ,

I 1 (1 -1~12)lf~(z)l < (1 -Iz l 2) sup{(1 -Iwl2)21f~'(~,)l}lzl (1 -Iz lh)-2dr

wED

< sup{(1 -Iwl2)21f~'(w)l}. wED

On the other hand < 6 6 / ~ / . (1 -Iwl2)21~1 . . . .

(1 Iw12)2[f~'(~)l - _ li ~ a ~ : : .

So IIf311S < 6~. Now we are going to show that f l is in BMOA. To see this, we may assume that f l (0) = 0.

Let g be continuous on the closure of the unit disk, define

l(g) = ID g'(z)f~(z) log IzidA(z).

Now we try to show that the pairing l(g) can be extended to a hounded linear functional on H I.

If this is true, Fefferman duality theorem gives us that there is a function F in BMOA such that

t(g) = L~ q(e~~176176

If g is continuous on the closure of the unit disk, polarization of the Littlewood-Paley integral

formula (on page 235 of [12]), then yields

- f g ' ( z )~ l~ - /g'(z)F---V~l~

So f2 = F. Let g be in H 1. Assume that g is continuous on the closure of the unit disk and write # = G 2

for G 6 H2(OD). Fubinis' theorem implies

- s g'(z)f~(z)log IzldA(z) =

i L - Iwl2)f'(w) - 3 g '(z)(1 (1 - - Z~) 3 l~ = <(1)

L - - " g'(z) loglz]dA(z)dA(w). / - 3 (I - lwl~)/'(w) -- (I -- -~-~)3 ,(:) . 1

Applying the Littlewood-Paley integral formula to inner integration, we have

g'(z) loglzldA(w)l = I j~,(:)(l - lwl')/'-rC;-~ / (i _- ~) .


I faA.O( 1 - Iwl2)fW~( (g -wg(0))),(w)dA(w) I

< C fa,(i)[G(w)G'(w)ldA(w)

_< C [ f a , u ~ Iaw ~' dA(w) ll/2r [ ( )l ~ ~ IG'(w)12( 1 - Iw]2)dA(w)] 1/:k - i I J a , ( / )

aA(~) By the Littlewood-Paley integral formula, the second factor is less than Hglll. Since Xo,(/) 1_1~ol 2

is a Carleson measure, we have

[Zo') -< Cllalln..

Then

I / g'(z)~O v) log IwldA(w)l ~ CIIgllm,

So the pairing l(g) is bounded on a dense subset of H 1. It extends a bounded linear functional on dA-A-AIzL is a Carleson measure, which implies that H 1. Thus diStB(f, BMOA) _< 6E for e if Xo,(/)O_lzl 2)

dist15(f , BMOA) is bounded by a multiple of quantity (B). If quanti ty (B) > quanti ty (A), there are two positive constants e and el and a function f,a in

BMOA so that dA(z)

xa ,u ) (! - I,I 2) is not a Carleson measure , e > s and

[If - s lib ~ q -

Since

we have

So

(1 -IzlZ)ls > (1 - I z l 2 ) l f ' ( z ) l - I I f - L~IIB,

f 4 , ( f ) C f~ . . . . (f*l)"

dA(z) < (1 -Iz[2)lf[,(z)12dm(z)

X~,,C:> (] _ izl 2) - (~ _ e~).

This gives us that Xn,x(l) ~(1-1zl 2) is a Carleson measure, which is a contradiction. Q.E.D.

As a consequence of Jones's distance formula, for a Bloch function f, f is in the closure of BMOA

in the Bloch space if and only if (1 - Izl2)-lXn,(/)dA is a Carleson measure for any r > 0. An interpolating sequence is a sequence {zn} in D with the property that for any bounded

sequence of complex number {An} there exists a function f in H ~176 such that f(zn) = An for all n. A well-known theorem of Carleson [10] states that a sequence {z~} is interpolating iff

inf H I z m - z ~ I > 0 . 1 - z , , -~zn

m~n


This condition is equivalent to the following covditions:

(1){zn} is separated;

(2 )~ (1 -Izn[)5,~ is a Carleson measure. A Blaschke product

~(~) = I I Iz~l ~ - z 7Z

is called an interpolating Blaschke product if its zero set {zn) is an interpolating sequence.

For a point z in D and a positive s < 1, the pseudohyperbolic disk D(z,s) with center z and

radius s is defined by Z - - W

D(z ,~ ) : { ~ D : ~ W < *}.

The following result is an extension of a well-known fact but we list it to make our argument

complete.

P r o p o s i t i o n 1. If {w,~} is an interpolating sequence, then for any positive s,

d# (1 2 -~ = - [z[ ) XUT=~D( . . . . )dA

is a Carleson measure.

P roof . Using the conformally invariant character of Carleson measures , we only need to prove

that oo sup ~ / ( 1 - [Wo]2)(1 _ [z[2)_,dA(z) < oc.

woeD~==l JD( . . . . . ) [1 Ydoz l"

Replacing z by its image Cz~(Z) = z-z, 1 - ~ , we reduce the integral on the pseudo-hyperbolic disk

D(wn, s) to one on the Euclidean disk D(0, ~). Making a change variable of w = Cz~ (z) implies

/D( . . . . , (1-[w~ -- ~0z[ 2 /O(0,~) ([ 1 -- (--1 ---- [ z ~ ) ( 1 -- [ w ~ -- ~-~nw0[2 [1 -- ~"~ w['dA(w~"

s 2 (1 -Iz,~]2)(l -]wol 2) < - ( 1 - s ) 2 [ 1 - T~w0[ 2

So sup / ( l - [ w o t 2 ) d # ( z ) /D ( 1 - I w e l 2)

woED JD [] .~oZl 2 --< CwoEDSlIp [1 W0Z[ 2 dl / ( z ) ,

where

dv is a Carleson measure since {z~} is an interpolating sequence and it follows that d# is a Carleson measure. Q.E.D.

M a x i m a l ideal space of H ~176 As in [15], the maximal ideal space of H a' is defined to be the space .hA of multiplicative

linear maps from H ~176 onto the field of complex numbers. The Gelfand theory represents H ~176 as

a suhalgebra of U=C(A,4) , the algebra of continuous, complex-valued function on .A.4 [16]. The


topology given by .Ad in the Gelfand theory is the weak-star topology, which makes i t a compact

Hausdorff space. If z is a point in the uni t disk D, then point evaluat ion at z is a mult ipl icat ive

l inear funct ional on H ~176 and so we can think of z as an element of AA. Thus we will freely th ink

of the uni t disk D as a subset of Ad. Carleson's Corona Theorem [11] s tates t h a t D is dense in

. ~ .

For points z and w in D, the pseudo-hyperbolic dis tance from z and w is

p(z,w) = Ir

Pick's l e m m a [12] says t ha t for z and w in D and a non-cons tant H ~176 funct ion with norm not

exceeding 1,

p(f(z), f(w)) <_ p(z, w).

Taking points r and ~b in .,%,4 and extending p to AA• by

p(~,~b) = sup{p(f(~),f(~p)): f C H~176 < 1),

we can par t i t ion AA into equivalence classes known as Gleason par ts , calling z 2 and ~p equivalent

provided p(zp,r < 1. We denote the Gleason par t to which (p belongs by G(~o). Hoffman has shown

tha t the Gleason par ts of M are ei ther trivial par ts or analytic discs [14]. For l a t t e r case he [16]

constructed a bijective map L~(z), called the Hoffman map , from D onto G(~o) sending 0 to

such t ha t f o L~ is holomorphic for all f in H ~176 and r (z) converges pointwise to L~(z) provided

the net {za} converges to c 2.

P r o p o s i t i o n 2. If ft is a subset of D such tha t the closure of fl in M does not contain any

tr ivial Gleason par ts then there exists a sequence {z,~} _C D which is a finite union of interpolat ing

sequences and a positive nulnber r such t ha t f~ C_ [-J~--1 D(z~,r).

P r o o f . If m is in the closure of ft in .AA but not in D, it follows from Corollary (on page 85 of

[16]) t ha t there is an in terpola t ing Blaschke product bm with in terpola t ing zeros {zmj} such t ha t

b,~(m) = 0. We define

which is an open set in the Gelfand topology on

number 0 < rm < 1 such t ha t

o(b~,~,,,) N D c

AA. In addi t ion for a very small em there is a

j= l

(see [17, page 404]). Now we choose such small numbers em to construct an open covering

{O(b,n,em)) of the closure of ~ in A.4. Since the closure of ~ in AA is compact , there is a fi-

n i te covering {O(m,, e,~,)}Y 1 of the closure of ft in .Ad.

Let r = max{rm,). We are going to prove tha t {D(zm,j,r)} c o v e r s ~ except for a finite num-

ber of points. If not , we may choose an infinite sequence {xk) C ~ such t h a t xk is not in

U~=I Uj~176 D(zm,j,r) Suppose t ha t m0 is in the closure of {xk) in M . Then m0 is in some

O(bm,em,) Hence {xk}~O(bm,,em,)is infinite and so is {xk)~Uj~lD(z,~,oj,r), which is a

contradict ion. Thus we can find a finite union of in terpola t ing sequences {z~) such tha t ~t C

[.J~~176 D(z~, r). Q.E.D.


B M O A a n d P ( U ) .

Hoffman [15] proved that for functions f in H ~176 (1 - [zI2)f'(z) is in U. Recently Axler and Zhu

[6] showed that for analytic functions f , f E P(/g) if and only if (1 - [z[2)f'(z) is in U. It is easy to

see that the later condition on f implies that P(L/) is a closed subspace of the Bloch space. In this

section we will show that P(U) equals to the closure of BMOA in P(U). To summarize all results on

P(/.d) and the closure of BMOA in the Bloch space we state the following theorem. We emphasize

here that not all these results are ours.

T h e o r e m 1. Suppose fEB and for e > 0, ~q~(f) = {z E D,(1 - [zl2)[f'(z)l > e}. Then the

following are equivalent:

(1) f i s in the closm'e of BMOA in B;

(2) f is in P(U);

(3) (1 - [ z I e ) f ' ( z ) is in /g ;

(4) For every e > 0, (1 - I z l e ) - l X n , ( / ) d A is a Carleson measure.

P r o o f . Equivalence of (1) and (4) is a consequence of Jones' distance formula [2]. Axler and Zhu

[6] showed that (2) is equivalent to (3) . Here we are going to show that (1) is equivalent to (2).

As pointed out earlier, BMOA is a subset of P(Lt). So the closure of BMOA in the Bloch space is a

subspace of P(/./). Hence we need only to show that every function f in P(/./) is also in the closure

of BMOA in 13. It follows fl'om Jones' distance formula that it is sufficient to show that for every

e > O, (1 - [zl2)-lx~{/}dA is a Carleson measure.

First we claim that the closure of f l , ( f ) in AA doesn't contain any trivial parts. If this is not

true, let m be a trivia] part in the closure. It follows from [17, 12, page 410] that there is a sequence

{z,z} in D such that a subnet {z~} of {z,~} converges to m. On the other hand Hoffman [16] also

proved that the M6bius maps Cz~ (z) converges to the Hoffman map Lm(z) pointwise. So we have

((1 - IzlZ)f ') o L,,,(w) = Ii2nm((1 - tzI2)f ') o Cz, (w)

for w in D. However since the Gleason part G(m) is trivial it follows from Theorem 17 in [6] that

(a -Izl2)f'(z) is zero on G(m). Hence, in particular

0 = lira ((1 - M 2 ) f ') o r = lira ((1 - [ z ~ ] a ) f ' ( z ~ ) ) .

This contradicts to

( 1 - Iz~l')lf" '(z~)l > ~.

Since the closure of f/~(f) in A,4. doesn't contain any trivia] parts, it follows from Proposition 2

that there are a finite union of interpolating sequences {z,~} and a positive number r < 1 such that

fl~(f) C 0 D(z,~,r). n = l

So

where

D - [ w [ 2 * -1 [ 1--[Wl 2 (1 - [ z l ' ) X~(I)dA < ]1) d# I1 ~ 1 ~ - I1 ~ 1 ~

d# = (1 - ]z l2)-IXUT=, D( .... )dA. Proposition 1 tells us that (1 - Izl2)-lxa~{I)dA is a Carleson measure. So is (1 - ]zl2)-lXn,(I)dA. Q.E.D.


Conformal maps and P(L/). Let f be a univalent function mapping D onto a simply connected domain fL For E _ 011,

f - l ( E ) = {~ e OD, f(~) exists (as a nontangential limit) and �9 E}. The harmonic measure w

evaluated at f (0) is given by w(E) = I f - l ( E ) l . If h is an increasing and continuous function on

R+ with h(0) = 0 and A(zo,r) = {z, ]z--zo] < r}, the Hausdorffmeasure Ah is defined for a Borel

set E by Ah(E) = } i ~ { i n f E h ( r j ) , E C U A ( z j , r j ) , r j < e}. Recently Makarov has found a J

beautiful estimate on radial growth of the Bloch function in the form of the law iterated logarithm

as follows for almost all 0,

where

/log log

1 1 = logloglog 1 - l~ 1 - r r"

As application of the estimate, Makarov showed that there exists a universal constant c > 0 such

that for any Jordan domain f / t h e harmonic measure on Of/ is absolutely continuous with respect

to the Hausdorff measure Ah~ where

he(t) = t exp{cx/ logt -1 log 3 t - l } .

Now we can state our main result in the section as follows.

T h e o r e m 2. If f is in P(L/), then

lim If(rei~ = 0 0 r--,, ~/log ~ l oga

a . e ,

P r o o f . If f is a Bloch function, it follows from a theorem of Ahlfors-Weil, and elementary distortion

theorem that f is representable in the form f = a logg ' for some constant a and some univalent

function g that extends quasi-conformaily from D onto a simply connected domain fL

If f is in BMOA, Of/is rectifiable [24, page 201] and hence by a wen-known theorem of F. and M.

Riesz [21, page 767] the harmonic measure w is equivalent to the arc-length A1. So the harmonic

measure w is absolutely continuous with respect to Ah,. Since r = t is a logarithmico-exponential

function [18], it follows from a theorem [18] that

lim Ig'("e~~ > o. ~ 1 exp{-e~ / log 1 ~-~:. Ioga 1I-~}

Let f+ = -t-logg', then g' = exp{:hf+}. Thus there is a positive constant C such that for r

sufficiently close to 1, we have

exp{e og l~ 1 - r + Re f~:} > e c > 0


for some number C. So

r l ~ r 1 + R e f • >--C" e og I~ 1 - r

Since J l o g 11_-~log3 ~ goes to oo a s r - * 1- , we have u ~

-Re f+(rd e) �9 > lira

- ~-~1 ? log ~ log 3

It follows that

lim Inef(reie)l = O. r-ol C o g ~ log a

In addition since i f E BMOA using the same argument above we can also show that

lira IImf(rd~ = O. r..-+l ? l o g 1 @ r log 3

If g is in P(lg), it follows from Theorem 1 that for any e > 0 there is a function f~ in BMOA

such that

IIg - f J I B < ~.

On the other hand the law of iterated logarithms of Makarov on the Bloch functions tells us that

for almost all O,

So

lim ]g(rel~ - L(rei~ < e. /log log -

li-m [~(rei~ < E, v/log log 1 -

Q.E.D. As defined in [17], a simply connected domain ~ is a 6-Makarov domain for ~5 > 0 if there is a

subset E of fl such that Ahs(E) = 0 but ~ (E) = 1.

One interesting part of [6] is the proof that P(U)#/3 via a concrete example. In this section we

use the geometric property outlined in the previous theorem to exhibit a class of Bloch functions

that do not belong to P(U).

C o r o l l a r y 1. Suppose that g : D ~ Q maps D conformally onto ft. If f = logg t is in P(H),

then whenever 6 > 0, fl is not 6 - Makarov don min.

P r o o f . Suppose f = logg' ; f 6 P(L/). It follows from Theorem 2 that

li~---~ If(rda)] = 0 a.e.O. v/log 1,__; l


So for 5 > O, let h~(t) = t exp{571og ~ log 3 !} and r t -Xhj l ( t ) ,'. e x p { - ~ 7 1 o g } log a ' = 7}. t

Thus

~.__.i ~ - - ~ - ~ ) > c l i m Ig'(rei~ limexp{Re f + 6 log _ log a l_--~}

1 = e lira e x p { , / l o g - - log 3 I - -L- ( Re f(re i~ v 1 - r 1 - ~ (log,!--rlog~ ~ + :))

= iX5.

So It follows from Theorem 1 [19] that w 3_ A W. Q.E.D.

Based on the result of Przytycki, Urbanski and Zdunik [24] on Julia sets we have the following

corollary which gives a geometric method to construct many Bloch function not in P(M).

C o r o l l a r y 2. Suppose that g is a conformal map of D onto A~o = domain of attraction at co

for the polynomial z 2 + c and z 2 + e acts hyperbolically on its Julia set. If f = alogg', f ~_ P(U).

At this point, it is natural to look at the space

.4 = {.f e B, lim,._~l.f(rd~ 11_---; log3 ~ = 0}. P r o p o s i t i o n 3. A is a M6bius-i)variant, closed subspace of B which contains BMOA.

P r o o f . The law of iterated logarithms of the Bloch functions implies that A is a closed subspace of

the Bloch space. In addition Theorem 2 tells us that A contains BMOA. It remains to prove that

,4 is M5bius-invariant. This follows from the following calculation. For A 6 D, let z = 9:~(re i~ =

pe i~. Then

1 - r 2 -- (I - Iz12)(1 - IAI ~-) _ / ~ r ~. I 1 - Xzl ~ ~ o 1 ---- 1 - - I , ~ i 2 1 -

Then

S O

Thus

I(f o ~:,)(rd~ If(pein)[ 41~ ~ l~ ~ 41~ ~ l~ x--"~

l i ~ I ( fe~, ) ( r~~ = lira If(Pdn)l = 0 a.e. ~. ,---+a v/log ~ log 3 11___.7 ~,-, 7log ~ log 3

~Ln]l( 1/71 ~ ' =0 ~.~.o. f o ~ ) ( r e i~ og _ l ~

Q . E . D .

It is natural to ask whether P(U) is a proper subspace of A . In next section, we will study the

Taylor coefficients of the Bloch functions to answer the question.

T a y l o r coef f ic ien ts .

The following proposition gives a condition on the Taylor coefficients of the Bloch functions

which are in P(U). Using the condition we can easily tell which gap series are in P(U). For a fixed

A in D and any anMytic function f, f o r has the Taylor expansion as follows:

c o

f o r = ~ ~(~)~r. 0


P r o p o s i t i o n 4. If f is in P(U), then for any fixed A in D, a(A)n ~ 0 as n --* oo.

P roof . Since P(/d) is the closure of BMOA in the Bloeh space and BMOA is Mhbius invariant,

P(U) is M6bins invariant. So it is sufficient to show that an ~ O. On the other hand [an[ ~_ 2[[fI[ B [3] and BMOA is a subset of H 2. So it follows from square summable Taylor coefficients of H 2 that

an ---> O as n -'-+ oo.

/D ( f~) (w) d ' " " We remark that simply differentiating under the integral ~ - ~ A[w) n times over and

using Hhlder's inequality yield that the Taylor coefficients a~ of P( f y ) goes to zero as n ~ oo

whenever f, g 6 H ~176

The following example in [9] shows that P(U) is a proper closed subspace of..4. For completeness

we list a few properties of the function they construct.

If by(z) = z 4J where j = 4 ~ and

b j ( z ) = 1 - 4 -(e+l) i~=4tbi(z) z2; j = 4 e + l , . . . , 4 e + l - 1 .

Let f ( z ) = anz n = ~ bk(z). If wn is the polynomial kernel defined by wo(z) = 1 + z and for n = 4 k = l

n > 0, ~ ( 4 n) - 1, ~bn - 0 outside (4n-1,4 ~+1) and linear on [4n-1,4 n] and on [4n,4~+1], and

bk(z) = f * wk then the computation in [9] shows that

l i , n = 0

and hence by [24], we have f cA . However a 4' = 1 whenever j = 4 ~. So by Proposition 4, f r

P(/4). This gives us two conditions that a function in P(/A) must satisfy, neither of which is sufficient separately. This leaves us with the open question of finding sufficient conditions on functions in ,4

which would pl~ce them in P(/4).

T h e l i t t l e B loch space a n d C O P .

As in [5], we use the Hoffman maps Lm(z) to define COP (constant on parts) and AOP (anulytic

on parts ) as follows:

A OP = { f E U, f o L,~ E H ~176 V m E A 4 / D }

and

C O P = { f 6 Ll, f o L,n i~ constant V m E A A / D } .

I f f ~ C(A4) then the Hankel operator Hf and the small Hankel operator HS are defined as follows:

H I : L~ --* (I - P)L2(D), Hl(g ) = (I - P) ( fg ) and HS: L~ ~ L~, Arl(g ) = P( fg ) where P is the

orthogonal projection of L 2 --2 onto L~. It is known that H I is compact if and only if f 6 A O P and

H I and H 7 are compact if any only if f 6 COP [27]. In addition, it is proved in [4] that if f 6 L~

t h e n / / 7 is compact if and only if f 6B0 . Now we state an analogue of Theorem 1.

T h e o r e m 3. For f 6B, the following are equivalent


(i) f 6 P(COP); (ii) (1 - I z l 2 ) f ' ( z ) 6 COP;

(iii) f E/N0; (iv) If for e > 0, a~(f ) = {z 6 D, (1 - [z[2)lf'(z)[ >_ e} then (1 - IzlZ)-lXn,(s)dA is a vanishing

Carleson measure for every e > 0.

P roof . Suppose (i) holds. Let f = P(g) where g e COP. It is easy to check that H7 = Hg" By

[27] Hy is compact and hence so is Hg = Hy. By Theorem 4 in [8] it follows that f E/N0; i.e., (iii) holds.

Now suppose that f 6 B0. Then (1 - Iz]2)f'(z) 6 Co(-D) C COP. So (ii) holds.

Suppose (ii) holds. We may assume f(0) = f ' (0) = 0. If (1 - I z l 2 ) f ' ( z ) 6 COP then so is

D 1 dA(w~ P(CO P), i.e., )f '(w)

(1 - I z l 2 ) f ' ( z ) / - 2 . By [6], f (z) = N(1- - z~) 2 j. Hence f 6 (i) holds.

Now we are going to prove that (iii) is equivalent to (iv). Suppose f E/N0. We claim that

whenever e > 0, f/~(f) = {z E D,(1 - [z l2) l f ' (z) [ >__ E} is a compact set. To prove the claim choose

a polynomial g such that Ilf - gl[/3 < e/2. Consequently f~ ( f ) c f~e/2(g) c {z 6 D, dist(z, OD) >_ ~/llg'lloo}, which is a compact set. Hence

1 - IAI 2 ( 1 - Izl2)-ldA(z) < C ( 1 - IA] ~) ~ dA(z) ~(/) I 1 "Azl ~ - I_<,<1 ]l--~zzI2 "-+ 0 as [A I --+ 1.

Conversely, suppose (1 - [zi2)-lxa,(s)dA is a vanishing Carleson measure V r > 0. If f ~B0 then

we may choose e > 0 such that lim (1 - Izi2)if'(z)I _> 2e and hence an interpolating sequence

{z~} _C D such that (1 - Iz~12)If'(z~)] > e V n. In particular, d# = E ( 1 - ] z ~ t 2 ) ~ . is a Carleson

measure. If 25 = inf~#mp(z,~,z,,) then the nniform continuity of (1 - I z l 2 ) f ' ( z ) with respect to oo

the pseudo-hyperbolic metric p allows us to choose r, 0 < r < 5 such that ~E/2(f) D_ U D(z~,r). n = l

These discs are pairwise disjoint and with r fixed, the Euclidean radius of D(z~,r) is comparable

t o (1 - I z ~ l " ) . H e n c e

(1 - I=,~1=)(1 - iz12) - , > [ ( 1 - Iz ,~l=)( ] - 1=12) - , dA(z) dA(z). ~/2 ]1 --2~zl 2 - JD( ..... ) 11 - ~ z l 2

Making a change variable w = Cz. (z) implies

/o /o (1 - Iz,~12)(] - Izl~) -~ dA(z) = l ~ d A ( w ) > ( . . . . ) 11 - ~,,zp (0,0 11 - ~ w l - 1 - r"

In particular - - ~ r2 l i m l ~ o l ~ ~ (1 - I z o l " ) ( l - Izl~) -~ dA(z) > - -

Cr- l1 - g 0 z p - 1 - r '

Q.E.D.

Added in proof-- K Stroethoff and T. Wolff have informed us that since any function in BMOA

has radial limits a.e., our proof of Theorem 2 can be shortened by invoking Mak~rov's estimate on

the radial growth of a Bloch function. We thank them both for their kind interest.


R e f e r e n c e s

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[22] Ch. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), 199-208.

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Department of Mathematics Cleveland State University Cleveland, OH 44115

and

Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794 Current address MSRI 1000 Centennial Drive Berkeley, Ca 94720

Mathematics Subject Classification: Primary 46E15, 46J 15

Submitted: January 18, 1993

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Analytic functions of bounded mean oscillation and the Bloch space