Research ArticleOn Distance Function in Some New Analytic Bergman TypeSpaces in C119899

Romi F Shamoyan1 and Olivera R MihiT2

1 Bryansk University 241050 Bryansk Russia2 Fakultet Organizacionih Nauka Jove Ilica 154 1100 Belgrade Serbia

Correspondence should be addressed to Olivera R Mihic oliveradjfonrs

Received 2 February 2014 Accepted 23 March 2014 Published 6 May 2014

Academic Editor Kehe Zhu

Copyright copy 2014 R F Shamoyan and O R Mihic This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new analyticBergman type spaces in higher dimensions in C119899 Also we study the same problem in anisotropic mixed norm ℎ(119901 119902 119904) spacesconsisting of 119899-harmonic functions on the unit polydisc of C119899

1 Introduction and Preliminaries

The goal of this paper is to add several new results fordistances in analytic Bergman type and Bergman type n-harmonic function spaces of functions of several variables

It turns out that some of our distance theorem whichwe proved before for the case of unit disk are valid undercertain natural conditions in various domains and variousBergman type analytic spaces Namely we look at analyticBergman spaces in minimal ball and so-called analytic 119870-Bergman spaces and in analytic Bergman-Orlicz spacesThese analytic spaces act as direct extensions of well-knownanalytic Bergman spaces in the unit disk These analyticspaces are relatively new and we will include some basic factson them in our paper they will also be needed for proofsof our assertions partially We will turn also to polydisk and119899-harmonic function spaces on them providing also a sharpresult there Note in less general case namely in analyticBergman type spaces in polydisk this result is already known(see [1 2])

Our intention in this paper is the same as our previouspapers on this topic namely we collect some facts fromearlier investigation concerning Bergman projection andBergman kernel and use them for our purposes in estimatesof dist

119884(119891X) function (distance function)

Following our previous paper [1 2] we easily see thatto obtain a sharp result for distance function we only needseveral tools and the scheme here is as follows First we needan embedding of our quazinormed analytic space (in anydomain) into another one (119883 sub 119884) this immediately posesa problem of dist

119884(119891119883) = inf

119892isin119883119891 minus 119892

119884for all 119891 isin 119884 119883

and then we need the Bergman reproducing formula for all 119891functions from119884 spaceThen finally we use the boundednessof Bergman type projections with |119870(119911 119908)| positive kernelacting from 119883 to 119883 together with Forelli-Rudin type sharpestimates of Bergman kernel These three tools were used ingeneral Siegel domain of second type polydisk and unit ballin [1ndash4] (see also various references there) We continue touse these tools providing new sharp (and not sharp) resultsin various spaces of analytic functions in this paper

First we provide a known result in the unit disk withcomplete proof taken from our previous paper ([1 2]) In theunit disk case all arguments here are short and transparentand are based on several tools like Forelli-Rudin type estimateand estimates for Bergman type projections with positiveBergman kernel and then we will see arguing similarly asin unit disk we will complete the proof of more complicatedcases easilyThe complete formulation of our theoremswill begiven but only sketches of proofs will be added and details ofproofs of higher dimensional cases will be left to readers

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 275416 10 pageshttpdxdoiorg1011552014275416

2 Journal of Function Spaces

Note that it is easy to see that our assertions may havevarious applications in approximation theory for example

We will need various definitions for formulations of mainresults All facts about polydisk are taken from [5]

Let 119899 isin N and C119899

= 119911 = (1199111 119911

119899) 119911

119896isin C 1 le 119896 le

119899 be the 119899-dimensional space of complex coordinates Wedenote the unit polydisk by

U119899

= 119911 isin C119899 1003816100381610038161003816z119896

1003816100381610038161003816lt 1 1 le 119896 le 119899 (1)

and the distinguished boundary of U119899 by

119879119899

= 119911 isin C119899 1003816100381610038161003816119911119896

1003816100381610038161003816= 1 1 le 119896 le 119899 (2)

By 1198982119899

we denote the volume measure on U119899 and by 119898119899we

denote the normalized Lebesgue measure on 119879119899 Let 119867(U119899

)

be the space of all holomorphic functions on U119899Throughout the paper we write 119862 (sometimes with

indexes) to denote a positive constant which might be differ-ent at each occurrence (even in a chain of inequalities) but isindependent of the functions or variables being discussed

Thenotation119860 ≍ 119861means that there is a positive constant119862 such that119861119862 le 119860 le 119862119861Wewill write for two expressions119860 ≲ 119861 if there is a positive constant 119862 such that 119860 lt 119862119861

We denote the interval (0 1) by 119868 accordingly (0 1)119899 by119868119899

The Hardy spaces denoted by 119867119901

(U119899

) (0 lt 119901 le infin) aredefined by 119867119901

(U119899

) = 119891 isin 119867(U119899

) sup0le119903lt1119872

119901(119891 119903) lt infin

where

119872119901

119901(119891 119903) = int

119879119899

1003816100381610038161003816119891 (119903120585)

1003816100381610038161003816

119901

119889119898119899(120585)

119872infin(119891 119903) = max

120585isin119879119899

1003816100381610038161003816119891 (119903120585)

1003816100381610038161003816

119903 isin 119868 119891 isin 119867 (U119899

)

(3)

As usual we denote by the vector (1205721 120572

119899)

For 120572119895gt minus1 119895 = 1 119899 0 lt 119901 lt infin recall that the

weighted Bergman space119860119901(U119899

) consists of all holomorphicfunctions on the polydisk such that

10038171003817100381710038171198911003817100381710038171003817

119901

119860119901

= int

U1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

119899

prod

119894=1

(1 minus1003816100381610038161003816119911119894

1003816100381610038161003816

2

)

120572119894

1198891198982119899(119911) lt infin (4)

When 1205721= sdot sdot sdot = 120572

119899= 120572 then we use notation 119860119901

120572(U119899

)If 119906 is 119899-harmonic (harmonic by each variable) then as

usual

119906 (11990311198901198941205931

119903119899119890119894120593119899

) =

infin

sum

1198961119896119899=minusinfin

1198621198961119896119899

119899

prod

119895=1

119903

|119896119895|

119895119890119894119896119895120593119895

(5)

Let further

ℎ119901

() = 119906 is 119899-harmonic

int

U119899

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896 1003816100381610038161003816119906 (119911

1 119911

119899)1003816100381610038161003816

119901

1198891198982119899(119911) lt infin

0 lt 119901 lt infin 120572119896gt minus1 119896 = 1 119899

(6)

Also we denote by B119899the unit ball in C119899

B119899= 119908 isin

C119899 |119908| lt 1 For 1 le 119901 lt +infin and 120572 gt minus1 denote by Hp120572(Bn)

(or Ap120572(Bn)) the space of all functions f holomorphic in Bn

satisfying the condition

int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

(1 minus |119908|2

)

120572

119889] (119908) lt +infin (7)

where 119889] is the Lebesgue measure in C119899

equiv R2119899 (see [6])Further for a complex number 120573 with Re120573 gt minus1 put

119888119899(120573) =

Γ (119899 + 1 + 120573)

120587119899sdot Γ (1 + 120573)

(8)

Let 119860infin120572(B

119899) = 119891 isin 119867(B

119899) sup

119911isinB119899

|119891(119911)|(1 minus |119911|)120572

lt

infin 120572 gt 0

Remark 1 Analytic119860infin120572spaces are well known in literature as

so-called growth spaces (see eg [5]) It can be shown easilythat these spaces are Banach spaces These spaces are playinga vital role in this paper and are embedded in Bergmanspaces 119860119901

120573 for large enough 120573 Hence the representation (10)

with 120573 large enough index depending on 120572 for all functionsfrom such classes is valid The mentioned embedding is wellknown and almost obvious and we leave the proof of it tointerested readers This fact nevertheless is crucial for theproof of Theorems 18 and 20 below and actually serves asbase of both proofs along with well-known Forelli-Rudinestimates for unit ball (see [5])

We have the following theorem

Theorem A (see [6]) Assume that 1 le 119901 lt +infin 120572 gt minus1 andthat the complex number 120573 satisfies the condition

Re120573 ge 120572 119901 = 1

Re120573 gt 120572 + 1119901

minus 1 1 lt 119901 lt infin

(9)

Then each function 119891 isin 119860119901120572(B

119899) admits the following integral

representations

119891 (119911) = 119888119899(120573) sdot int

B119899

119891 (119908) (1 minus |119908|2

)

120573

(1 minus ⟨119911 119908⟩)119899+1+120573

119889] (119908) 119911 isin B119899

(10)

119891 (0) = 119888119899(120573) sdot int

B119899

119891 (119908)(1 minus |119908|2

)

120573

(1 minus ⟨119911 119908⟩)119899+1+120573

119889] (119908) 119911 isin B119899(11)

where ⟨sdot sdot⟩ is the Hermitian inner product in C119899

Journal of Function Spaces 3

For 119899 gt 1 the theorem was proved in [7] (when 120572 = 0)and in [8 9] (when 120572 gt minus1)

Inmonographs [5] one can find numerous applications ofthe formulas (10) and (11) in the complex analysis In [6] theauthor generalized Theorem A to the space with anisotropicweight function such as assuming that 1 le 119901 lt +infin 120572 =(120572

1 120572

119899) isin R119899 satisfies the conditions

120572119899gt minus1

120572119899+ 120572

119899minus1gt minus2

120572119899+ 120572

119899minus1+ 120572

119899minus2gt minus3

120572119899+ 120572

119899minus1+ 120572

119899minus2+ sdot sdot sdot + 120572

2gt minus (119899 minus 1)

120572119899+ 120572

119899minus1+ 120572

119899minus2+ sdot sdot sdot + 120572

2+ 120572

1gt minus119899

(12)

and then introduced the spaces 119860119901120572(B

119899) of functions 119891

holomorphic in B119899satisfying the condition

int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

lt +infin

(13)

For these anisotropic spaces similarities of the inte-gral representations (10) and (11) are obtained but in thementioned generalization special kernels 119878

12057311205732sdotsdotsdot120573119899

(119911 119908) equiv

119878120573(119911 119908) (where 120573

119894 1 le 119894 le 119899 are associated with 120572

119894 1 le

119894 le 119899 and 119901 in a special way) appear instead of 119888119899(120573) sdot (1 minus

⟨119911 119908⟩)minus(119899+1+120573) (see [6] Theorem 47)

Suppose now that 120572 = (1205721 120572

2 120572

119899) isin R119899 We put

119898120572= sum

120572119894lt0

minus 120572119894 119897

120572= sum

120572119894lt0

120572119894 (14)

We will write 120572 ≺ (lowast) only if the following conditions aresatisfied

120572119899+ 120572

119899minus1+ sdot sdot sdot + 120572

119894gt minus (119899 + 1 minus 119894) (1 le 119894 le 119899) (15)

Similarly if 120573 = (1205731 120573

2 120573

119899) isin C119899 then we will write 120573 ≺

(lowast) if only Re120573 equiv (Re1205731Re120573

2 Re120573

119899) ≺ (lowast)

We have the following lemmas

Lemma 2 (see [6]) If 120572 ≺ (lowast) then

int

B119899

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

=

120587119899

prod119899

119894=1(120572

119899+ 120572

119899minus1+ sdot sdot sdot + 120572

119894+ 119899 + 1 minus 119894)

lt +infin

(16)

We need also heavily the well-known Forelli-Rudin typeestimates for unit ball (see [5])

Proposition 3 Suppose 119888 gt 0 is real and 119905 gt minus1 Then theintegral

119869119888119905(119911) = int

B119899

(1 minus |119908|2

)

119905

119889] (119908)

|1 minus ⟨119911 119908⟩|119899+1+119905+119888

119911 isin B119899

(17)

has the following asymptotic property

119869119888119905sim (1 minus |119911|

2

)

minus119888

as |119911| 997888rarr 1 minus (18)

Definition 4 Assume that 0 lt 119901 lt +infin 120572 = (1205721 120572

119899) ≺

(lowast) Denote by 119871119901120572(B

119899) the space of all complex-valued

functions 119891 in B119899with

119901

120572(119891) equiv int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

times

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

lt infin

(19)

We obviously have

119901

120572(119891 + 119892) le 2

119901

(119901

120572(119891) +

119901

120572(119892))

119901

120572(119888119891) = 119888

119901

119901

120572(119891)

(20)

Lemma 5 (see [6]) Assume that 0 lt 119901 lt +infin 120572 ≺ (lowast) and119891 isin 119867(B

119899) Then

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

le

22119899+119897120572

119888119899sdot (1 minus |119911|)

2119899+119897120572

sdot 119901

120572(119891) forall119911 isin B

119899 (21)

where 119888119899= ](B

119899) is the volume of the unit ball of C119899

Note that this lemma leads to an embedding theoremimmediately which allows us to pose a distance problem asunit disk case

Corollary 6 (see [6]) 119867119901

120572(B

119899) is a closed subspace in 119871119901

120572(B

119899)

We will need some facts from theory of Bergman-Orliczspaces (see [10 11])

Let now Ψ [0infin) rarr [0infin) be a continuous non-decreasing function which vanishes and is continuous at 0Given a probabilistic space (ΩP) we define the Orlicz class119871Ω

(ΩP) as the set of all (equivalence classes of) measurablefunctions 119891 on Ω such that int

Ω

Ψ(|119891|119862)119889P lt infin for some0 lt 119862 lt infin We used to define the Morse-Transue space119872

Ψ

(ΩP) by

119872Ψ

(ΩP) = 119891Ω 997888rarr Cmeasurable

int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

119862

)119889P lt infin for any 119862 gt 0

(22)

4 Journal of Function Spaces

and we also introduce the following set

LΨ

(ΩP)

= 119891Ω 997888rarr Cmeasurable intΩ

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P lt infin

(23)

These three sets are not vector spaces and do not coincidebut we have

119872Ψ

(ΩP) subLΨ

(ΩP) sub 119871Ψ

(ΩP) (24)

We also define the Luxembourg gauge on 119871Ψ(ΩP) by

10038171003817100381710038171198911003817100381710038171003817Ψ= inf (120582 gt 0 int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

120582

) 119889P le 1) (25)

This functional is homogeneous and is 0 if and only if 119891 = 0but it is not subadditive

We say that the two functions Ψ1and Ψ

2as above are

equivalent if there exists some constant 119862 such that

119862Ψ1(119888119909) le Ψ

2(119909) le 119862

minus1

Ψ1(119862

minus1

119909) (26)

for any 119909 large enough Two equivalent functions define thesame Orlicz class with equivalent Luxembourg functionals

In order to define a good topology on 119871Ψ(ΩP) and to getproperties convenient for our purpose we will assume thatΨsatisfies the following definition

Definition 7 Let 0 lt 119901 le 1 One says that Ψ [0infin) rarr[0infin) is a growth function of order 119901 if it satisfies the twofollowing conditions

(1) Ψ is of lower type 119901 that is Ψ(119910119909) le 119910119901Ψ(119909) for any0 lt 119910 le 1 and at least for 119909 large enough

(2) 119909 997891rarr Ψ(119909)119909 is nonincreasing at least for every 119909large enough

We will say that Ψ is a growth function if it is a growthfunction of order 119901 for some 0 lt 119901 le 1

Sometimes we will use lower 119901 index for our growthfunction below

In particular a growth function Ψ is equivalent to thefunction 119909 997891rarr int119909

0

(Ψ(119904)119904)119889119904which is concave (see [12]) Nowfor such a concave growth function of order 119901

1198711

(ΩP) sub 119872Ψ

(ΩP) =LΨ

(ΩP) = 119871Ψ

sub 119871119901

(ΩP)

10038171003817100381710038171198911003817100381710038171003817Ψ≲ minint

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P (int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P)

119901

(27)

while

10038171003817100381710038171198911003817100381710038171003817Ψ= int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P ≲ max 100381710038171003817

10038171198911003817100381710038171003817Ψ10038171003817100381710038171198911003817100381710038171003817

119901

Ψ (28)

Moreover if we define 119889Ψ(119891 119892) = 119891 minus 119892

Ψ(119889Ψ(119891 119892) =

119891 minus 119892Ψ) then 119889

Ψand 119889Ψ (that we simply denote by sdot

Ψ

and sdot Ψ) are two equivalent metrics on 119871Ψ(ΩP) for whichit is completeWithout loss of generality we then assume thatevery growth function thatwe consider further is concave anddiffeomorphic

When we deal with spaces of holomorphic functionsit is very natural to require subharmonicity Then we willassume that Ψ is such that Ψ(|119891|) is subharmonic when119891 is holomorphic We will refer to such a function as asubharmonic-preserving function

Here are some examples of (concave) growth functionthat we may consider further

For Ψ a subharmonic-preserving growth function and120572 gt minus1 the weighted Bergman-Orlicz space 119860Ψ

120572of the ball

consists of those holomorphic functions onB119899which belongs

to the Orlicz space 119871Ψ(B119899 ]120572) where as usual for any real

number 120572 119889]120572(119911) = 119888

120572(1 minus |119911|

2

)120572

119889](119911) for |119911| lt 1 Here if120572 le minus1 119888

120572= 1 and if 120572 gt minus1 119888

120572= Γ(119899+1+120572)Γ(119899+1)Γ(120572+1)

is the normalizing constant so that ]120572has unit total mass

Further we will denote by sdot 120572Ψ

the correspond-ing Luxembourg (quasi)norm and by sdot Ψ

120572the quantity

intB119899

Ψ(|119891|)119889]120572 119860

Ψ

120572is metric space for the distance 119889

120572Ψor

119889Ψ

120572defined by respectively 119889

120572Ψ(119891 119892) = 119891 minus 119892

120572Ψand

119889Ψ

120572(119891 119892) = 119891 minus 119892

Ψ

120572 If Ψ(119905) = 119905119901 then we recover the

usual weighted Bergman space 119860119901120572 We have the following

inclusions

1198601

120572sub 119860

Ψ

120572sub 119860

119901

120572 (29)

whenever Ψ is a growth function of order 119901Note that the second inclusion which we see before leads

to another inclusion namely 119860Ψ120572 is embedded in 119860infin

(120572+119899+1)119901

This last inclusion will lead us to distance theorem Weshowed a complete analogue of a theorem in the unit diskbelow

It is important to mention that a linear operator 119879 from119860Ψ1

120572to 119883 with 119883 = 119871Ψ2(B

119899 ]120572) or 119883 = 119860Ψ2

120572 where Ψ

1and Ψ

2

are two growth functions is continuous if and only if it mapsa bounded set into a bounded set or equivalently if and onlyif there exists a constant 119862 gt 0 such that

max (1003817100381710038171003817119879 (119891)

1003817100381710038171003817120572Ψ2

1003817100381710038171003817119879 (119891)

1003817100381710038171003817

Ψ2

120572) le 119862 (30)

for any 119891 isin 119860Ψ1120572such that min(119891

120572Ψ1

119891Ψ1

120572) le 1 If119883 = C

then119879 is bounded if and only if |119879(119891)| le 119862 for119891 as previouslyThe next proposition says that the point evaluation

functionals are continuous on 119860Ψ120572

Proposition 8 (see [11]) Let 120572 gt minus1 and let Ψ be asubharmonic-preserving growth function For any 120572 isin B

119899and

any 119891 isin 119860Ψ120572 one has

1003816100381610038161003816119891 (119886)

1003816100381610038161003816le Ψ

minus1

((

2

1 minus |119886|

)

(119899+120572+1)

)10038171003817100381710038171198911003817100381710038171003817120572Ψ (31)

The proof is the same as that of [10 Proposition 19] andso is omitted (still use the hypothesis that Ψ(|119891|) is subhar-monic) We easily deduce from this and the completeness of119871Ψ the following result

Journal of Function Spaces 5

Corollary 9 (see [11]) 119860Ψ120572 endowed with 119889

120572Ψor 119889Ψ

120572 is a

complete metric space

Some facts from theory of 119899-harmonic function spaceswill be needed by us We will use the same notation for119872

119901(119891 119903) for harmonic and analytic functions

Definition 10 The quasinormed space 119871(119901 119902 120572) (0 lt 119901 119902 leinfin 120572 = (120572

1 120572

119899) 120572

119895gt minus1 119895 = 1 119899 is the set of those

functions 119891(119911) measurable in the polydisc U119899 for which thequasinorm

10038171003817100381710038171198911003817100381710038171003817119901119902120572

=

(int

119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895119902minus1

119872119902

119901(119891 119903)

119899

prod

119895=1

119889119903119895)

1119902

0 lt 119902 lt infin

ess sup119903isin119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895

119872119901(119891 119903)

119902 = infin

(32)

is finite For the subspace of 119871(119901 119902 120572) consisting of 119899-harmonic functions let ℎ(119901 119902 120572) = ℎ(U119899

) cap 119871(119901 119902 120572) (see[13])

For 119901 = 119902 lt infin the spaces ℎ(119901 119902 120572) coincide with thewell-known weighted Bergman spaces while for 119902 = infin theyare known as growth spaces

Let now 1 le 119901 119902 le infin120572119895gt 0 The following inclusion

will be used by us (see [13])

ℎ (119901 119902 120572) sub ℎ (infininfin 120572) (33)

This inclusion as in unit disk case (see [2]) allows us topose a distance problem in spaces of 119899-harmonic functionswe consider in this paper (we also did the same in spaces ofanalytic functions in the unit polydisk (see [1]))

Now we define Poisson kernel 119875120572in the unit disk

119875120572(119911) = Γ (120572 + 1) [Re 2

(1 minus 119911)120572+1minus 1] 119911 isin U 120572 ge 0

(34)

Let 119875120572(119911 120585) = 119875

120572(119911120585) For polydisk we have

119875120572(119911 120585) =

119899

prod

119895=1

119875120572119895

(119911119895 120585119895) (35)

where 120572 = (1205721 120572

119899) 120572

119895ge 0 120585 isin U119899

119911 isin U119899 119875120572is

119899-harmonic by both variables 119911 and 120585 and obviously holds119875120572(119911 120585) = 119875

120572(120585 119911) = 119875

120572(119911 120585) (see [13])

Reproducing integral formula Poisson-Bergman type willbe presented in the following theorem

Theorem B (see [13]) Let 120572119895gt 0 119906 isin ℎ(119901 119902 120572) If 0 lt 119901

119902 le infin 120573119895gt max120572

119895+ 1119901 minus 1 120572

119895 or 1 le 119901 le infin 0 lt 119902 le 1

120573119895ge 120572

119895(1 le 119895 le 119899)

119906 (119911) =

1

prod119899

119895=1Γ (120573

119895)

timesint

U119899

119899

prod

119895=1

(1 minus10038161003816100381610038161205771003816100381610038161003816

2

)

120573119895minus1

119875120573(119911 120577) 119906 (120577) 119889119898

2119899(120577)

119911 isin U119899

(36)

Let 119878120573120582

be a linear operator Bergman type

119878120573120582(119891) (119911)

=

(1 minus |119911|2

)

120582

Γ (120573 + 120582)

times int

U119899(1 minus10038161003816100381610038161205851003816100381610038161003816

2

)

120573minus1 10038161003816100381610038161003816119875120573+120582(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(37)

Theorem C (see [13]) Let 1 le 119901 119902 le infin 120572119895 120573

119895 120582

119895isin

R 120573119895gt 120572

119895gt minus120582

119895 1 le 119895 le 119899 Then the operator 119878

120573120582

continuously maps the space 119871(119901 119902 120572) into itself Moreoverthen operator 119878

120573120582is bounded in 119871(119901 119902 120572) if and only if 120573

119895gt

120572119895gt minus120582

119895 1 le 119895 le 119899

We will also need basic facts from theory of Bergmanspaces in minimal ball (see [14 15]) For the convenience ofreaders we will give some details about them

We consider the domain Blowastin C119899

119899 ge 2 defined by

Blowast= 119911 isin C

119899 |119911|2 + |119911 ∙ 119911| lt 1 (38)

where

119911 ∙ 119908 =

119899

sum

119895=1

119911119895119908119895 (39)

for 119911 and 119908 in C119899 This is the unit ball of C119899 with respect tothe norm

119873lowast(119911) = radic|119911|

2

+ |119911 ∙ 119911| 119911 isin C119899

(40)

The norm119873 = 119873lowastradic2was introduced by Hahn and Pflug in

[16] where it was shown to be the smallest norm in C119899 thatextends the Euclidean norm inR119899 under certain restrictions

The domainBlowasthas since been studied by Hahn and Pflug

[16] Mengotti and Youssfi [15] and Oeljeklaus and Youssfi[17] In particular it is well known that the automorphismgroup of B

lowastis compact and its identity component is

Aut0O (Blowast) = 119878

1

sdot 119878119874 (119899R) (41)

where the 1198781-action is diagonal and the 119878119874(119899R)-action isby matrix multiplication (see [17] eg) It is also well knownthat B

lowastis a nonhomogeneous domain Its singular boundary

consists of all points 119911 with 119911 ∙ 119911 = 0 and the regular part ofthe boundary of B

lowastconsists of strictly pseudoconvex points

6 Journal of Function Spaces

For each 119904 gt minus1 we let V119904denote the measure on B

lowast

defined by

119889V119904(119911) = (1 minus 119873

2

lowast(119911))

119904

119889V (119911) (42)

where V denotes the normalized Lebesguemeasure onBlowast For

all 0 lt 119901 lt infin we define

A119901

119904(B

lowast) =H (B

lowast) cap 119871

119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) (43)

whereH(Blowast) is the space of all holomorphic functions onB

lowast

Naturally the spaces A119901

119904(B

lowast) are called weighted Bergman

spaces of Blowast

When 119901 = 2 there exists an orthogonal projection from1198712

(Blowast 119889V

119904) onto A2

119904(B

lowast) This will be called the weighted

Bergman projection and is denoted by P119904Blowast

It is well known that P

119904Blowast

is an integral operator on1198712

(Blowast 119889V

119904) namely

P119904Blowast

119891 (119911) = int

Blowast

K119904Blowast

(119911 119908) 119891 (119908) 119889V119904(119908) (44)

where

K119904Blowast

(119911 119908) =

A (119883 119884)(119899

2+ 119899 minus 119904) V

119904(B

lowast) (119883

2minus 119884)

119899+1+119904(45)

is the weighted Bergman kernel Here

119883 = 1 minus 119911 ∙ 119908 119884 = (119911 ∙ 119911)119908 ∙ 119908 (46)

and A(119883 119884) is the suminfin

sum

119895=0

119888119899119904119895119883119899+119904minus1minus2119895

119884119895

times [2 (119899 + 119904)119883 minus

(119899 minus 2119895 + 119904) (119899 + 1 + 2119904)

119899 + 119904 + 1

(1198832

minus 119884)]

(47)

As usual

119888119899119904119895= (

119899 + 119904 + 1

2119895 + 1)

=

(119899 + 119904 + 1) (119899 + 119904) sdot sdot sdot (119899 + 119904 minus 2119895 + 1)

(2119895 + 1)

(48)

is the binomial coefficient and V119904(B

lowast) is the weighted

Lebesgue volume of Blowast See [15 17] for these formulas and

more information about the weighted Bergman kernelsBecause of the infinite sum 119860(119883 119884) the formula for K

119904Blowast

is not really a closed form As such it is inconvenient for usto do estimates for P

119904Blowast

directlyThus we employ a techniquethat was used in [15] More specifically we relate the domainBlowastto the hypersurface M of the Euclidean unit ball in C119899+1

defined by

M = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 |119911| lt 1 (49)

If Pr C119899+1

rarr C119899 is defined by

Pr (1199111 119911

119899 119911

119899+1) = (119911

1 119911

119899) (50)

and F = Pr|M then F M rarr B

lowastminus 0 Let

H = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 (51)

It was proved in [17] that there is an 119878119874(119899 + 1C)-invariantholomorphic form 120572 on H Moreover this form is uniqueup to a multiplicative constant In fact after appropriatenormalization the restriction to H cap (C 0)119899+1 of this formis given by

120572 (119911) =

119899+1

sum

119895=1

(minus1)119895minus1

119911119895

1198891199111and sdot sdot sdot and

119889119911

119895and sdot sdot sdot and 119889119911

119899+1 (52)

Our norm estimates for the weighted Bergman projectionP119904Blowast

on 119871119901 spaces will be based on the correspondingestimates onM Therefore we will consider the spaces 119871119901

119904(M)

consisting of measurable complex-valued functions 119891 on M

such that

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119904(M)= int

M

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

(1 minus |119911|2

)

119904120572 (119911) and 120572 (119911)

119862

lt infin (53)

where

119862 = (minus1)119899(119899+1)2

(2119894)119899

(54)

We use 119860119901119904(M) to denote the subspace of all holomorphic

functions in 119871119901119904(M) The weighted Bergman projection P

119904M

is then the orthogonal projection from 1198712119904(M) onto 1198602

119904(M)

Again it is well known that P119904M is an integral operator on

1198712

119904(M) given by the formula

P119904M119891 (119911) = int

M

K119904M (119911 119908) 119891 (119908) (1 minus |119908|

2

)

119904120572 (119911) and 120572 (119911)

119862

(55)

where K119904M is the corresponding Bergman kernel

The starting point for our analysis is the following closedform of K

119904M which was obtained as Theorem 32 in [15]

Lemma 11 (see [14]) The weighted Bergman kernel K119904M of

1198602

119904(M) is given by

K119904M (119911 119908) =

119862 (119899 minus 1 + (119899 + 1 + 2119904) 119911 ∙ 119908)

(1 minus 119911 ∙ 119908)119899+119904+1

(56)

where 119862 is a certain constant that depends on 119899 and 119904

Let 119891Blowastrarr C be a measurable function We define a

function T119891 onM by

(T119891) (119911) =119911119899+1

(2(119899 + 1)2

)

1119901

(119891 ∘ F) (119911)

=

119911119899+1119891 (119911

1 119911

119899)

(2(119899 + 1)2

)

1119901

(57)

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

Note that it is easy to see that our assertions may havevarious applications in approximation theory for example

We will need various definitions for formulations of mainresults All facts about polydisk are taken from [5]

Let 119899 isin N and C119899

= 119911 = (1199111 119911

119899) 119911

119896isin C 1 le 119896 le

119899 be the 119899-dimensional space of complex coordinates Wedenote the unit polydisk by

U119899

= 119911 isin C119899 1003816100381610038161003816z119896

1003816100381610038161003816lt 1 1 le 119896 le 119899 (1)

and the distinguished boundary of U119899 by

119879119899

= 119911 isin C119899 1003816100381610038161003816119911119896

1003816100381610038161003816= 1 1 le 119896 le 119899 (2)

By 1198982119899

we denote the volume measure on U119899 and by 119898119899we

denote the normalized Lebesgue measure on 119879119899 Let 119867(U119899

)

be the space of all holomorphic functions on U119899Throughout the paper we write 119862 (sometimes with

indexes) to denote a positive constant which might be differ-ent at each occurrence (even in a chain of inequalities) but isindependent of the functions or variables being discussed

Thenotation119860 ≍ 119861means that there is a positive constant119862 such that119861119862 le 119860 le 119862119861Wewill write for two expressions119860 ≲ 119861 if there is a positive constant 119862 such that 119860 lt 119862119861

We denote the interval (0 1) by 119868 accordingly (0 1)119899 by119868119899

The Hardy spaces denoted by 119867119901

(U119899

) (0 lt 119901 le infin) aredefined by 119867119901

(U119899

) = 119891 isin 119867(U119899

) sup0le119903lt1119872

119901(119891 119903) lt infin

where

119872119901

119901(119891 119903) = int

119879119899

1003816100381610038161003816119891 (119903120585)

1003816100381610038161003816

119901

119889119898119899(120585)

119872infin(119891 119903) = max

120585isin119879119899

1003816100381610038161003816119891 (119903120585)

1003816100381610038161003816

119903 isin 119868 119891 isin 119867 (U119899

)

(3)

As usual we denote by the vector (1205721 120572

119899)

For 120572119895gt minus1 119895 = 1 119899 0 lt 119901 lt infin recall that the

weighted Bergman space119860119901(U119899

) consists of all holomorphicfunctions on the polydisk such that

10038171003817100381710038171198911003817100381710038171003817

119901

119860119901

= int

U1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

119899

prod

119894=1

(1 minus1003816100381610038161003816119911119894

1003816100381610038161003816

2

)

120572119894

1198891198982119899(119911) lt infin (4)

When 1205721= sdot sdot sdot = 120572

119899= 120572 then we use notation 119860119901

120572(U119899

)If 119906 is 119899-harmonic (harmonic by each variable) then as

usual

119906 (11990311198901198941205931

119903119899119890119894120593119899

) =

infin

sum

1198961119896119899=minusinfin

1198621198961119896119899

119899

prod

119895=1

119903

|119896119895|

119895119890119894119896119895120593119895

(5)

Let further

ℎ119901

() = 119906 is 119899-harmonic

int

U119899

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896 1003816100381610038161003816119906 (119911

1 119911

119899)1003816100381610038161003816

119901

1198891198982119899(119911) lt infin

0 lt 119901 lt infin 120572119896gt minus1 119896 = 1 119899

(6)

Also we denote by B119899the unit ball in C119899

B119899= 119908 isin

C119899 |119908| lt 1 For 1 le 119901 lt +infin and 120572 gt minus1 denote by Hp120572(Bn)

(or Ap120572(Bn)) the space of all functions f holomorphic in Bn

satisfying the condition

int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

(1 minus |119908|2

)

120572

119889] (119908) lt +infin (7)

where 119889] is the Lebesgue measure in C119899

equiv R2119899 (see [6])Further for a complex number 120573 with Re120573 gt minus1 put

119888119899(120573) =

Γ (119899 + 1 + 120573)

120587119899sdot Γ (1 + 120573)

(8)

Let 119860infin120572(B

119899) = 119891 isin 119867(B

119899) sup

119911isinB119899

|119891(119911)|(1 minus |119911|)120572

lt

infin 120572 gt 0

Remark 1 Analytic119860infin120572spaces are well known in literature as

so-called growth spaces (see eg [5]) It can be shown easilythat these spaces are Banach spaces These spaces are playinga vital role in this paper and are embedded in Bergmanspaces 119860119901

120573 for large enough 120573 Hence the representation (10)

with 120573 large enough index depending on 120572 for all functionsfrom such classes is valid The mentioned embedding is wellknown and almost obvious and we leave the proof of it tointerested readers This fact nevertheless is crucial for theproof of Theorems 18 and 20 below and actually serves asbase of both proofs along with well-known Forelli-Rudinestimates for unit ball (see [5])

We have the following theorem

Theorem A (see [6]) Assume that 1 le 119901 lt +infin 120572 gt minus1 andthat the complex number 120573 satisfies the condition

Re120573 ge 120572 119901 = 1

Re120573 gt 120572 + 1119901

minus 1 1 lt 119901 lt infin

(9)

Then each function 119891 isin 119860119901120572(B

119899) admits the following integral

representations

119891 (119911) = 119888119899(120573) sdot int

B119899

119891 (119908) (1 minus |119908|2

)

120573

(1 minus ⟨119911 119908⟩)119899+1+120573

119889] (119908) 119911 isin B119899

(10)

119891 (0) = 119888119899(120573) sdot int

B119899

119891 (119908)(1 minus |119908|2

)

120573

(1 minus ⟨119911 119908⟩)119899+1+120573

119889] (119908) 119911 isin B119899(11)

where ⟨sdot sdot⟩ is the Hermitian inner product in C119899

Journal of Function Spaces 3

For 119899 gt 1 the theorem was proved in [7] (when 120572 = 0)and in [8 9] (when 120572 gt minus1)

Inmonographs [5] one can find numerous applications ofthe formulas (10) and (11) in the complex analysis In [6] theauthor generalized Theorem A to the space with anisotropicweight function such as assuming that 1 le 119901 lt +infin 120572 =(120572

1 120572

119899) isin R119899 satisfies the conditions

120572119899gt minus1

120572119899+ 120572

119899minus1gt minus2

120572119899+ 120572

119899minus1+ 120572

119899minus2gt minus3

120572119899+ 120572

119899minus1+ 120572

119899minus2+ sdot sdot sdot + 120572

2gt minus (119899 minus 1)

120572119899+ 120572

119899minus1+ 120572

119899minus2+ sdot sdot sdot + 120572

2+ 120572

1gt minus119899

(12)

and then introduced the spaces 119860119901120572(B

119899) of functions 119891

holomorphic in B119899satisfying the condition

int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

lt +infin

(13)

For these anisotropic spaces similarities of the inte-gral representations (10) and (11) are obtained but in thementioned generalization special kernels 119878

12057311205732sdotsdotsdot120573119899

(119911 119908) equiv

119878120573(119911 119908) (where 120573

119894 1 le 119894 le 119899 are associated with 120572

119894 1 le

119894 le 119899 and 119901 in a special way) appear instead of 119888119899(120573) sdot (1 minus

⟨119911 119908⟩)minus(119899+1+120573) (see [6] Theorem 47)

Suppose now that 120572 = (1205721 120572

2 120572

119899) isin R119899 We put

119898120572= sum

120572119894lt0

minus 120572119894 119897

120572= sum

120572119894lt0

120572119894 (14)

We will write 120572 ≺ (lowast) only if the following conditions aresatisfied

120572119899+ 120572

119899minus1+ sdot sdot sdot + 120572

119894gt minus (119899 + 1 minus 119894) (1 le 119894 le 119899) (15)

Similarly if 120573 = (1205731 120573

2 120573

119899) isin C119899 then we will write 120573 ≺

(lowast) if only Re120573 equiv (Re1205731Re120573

2 Re120573

119899) ≺ (lowast)

We have the following lemmas

Lemma 2 (see [6]) If 120572 ≺ (lowast) then

int

B119899

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

=

120587119899

prod119899

119894=1(120572

119899+ 120572

119899minus1+ sdot sdot sdot + 120572

119894+ 119899 + 1 minus 119894)

lt +infin

(16)

We need also heavily the well-known Forelli-Rudin typeestimates for unit ball (see [5])

Proposition 3 Suppose 119888 gt 0 is real and 119905 gt minus1 Then theintegral

119869119888119905(119911) = int

B119899

(1 minus |119908|2

)

119905

119889] (119908)

|1 minus ⟨119911 119908⟩|119899+1+119905+119888

119911 isin B119899

(17)

has the following asymptotic property

119869119888119905sim (1 minus |119911|

2

)

minus119888

as |119911| 997888rarr 1 minus (18)

Definition 4 Assume that 0 lt 119901 lt +infin 120572 = (1205721 120572

119899) ≺

(lowast) Denote by 119871119901120572(B

119899) the space of all complex-valued

functions 119891 in B119899with

119901

120572(119891) equiv int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

times

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

lt infin

(19)

We obviously have

119901

120572(119891 + 119892) le 2

119901

(119901

120572(119891) +

119901

120572(119892))

119901

120572(119888119891) = 119888

119901

119901

120572(119891)

(20)

Lemma 5 (see [6]) Assume that 0 lt 119901 lt +infin 120572 ≺ (lowast) and119891 isin 119867(B

119899) Then

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

le

22119899+119897120572

119888119899sdot (1 minus |119911|)

2119899+119897120572

sdot 119901

120572(119891) forall119911 isin B

119899 (21)

where 119888119899= ](B

119899) is the volume of the unit ball of C119899

Note that this lemma leads to an embedding theoremimmediately which allows us to pose a distance problem asunit disk case

Corollary 6 (see [6]) 119867119901

120572(B

119899) is a closed subspace in 119871119901

120572(B

119899)

We will need some facts from theory of Bergman-Orliczspaces (see [10 11])

Let now Ψ [0infin) rarr [0infin) be a continuous non-decreasing function which vanishes and is continuous at 0Given a probabilistic space (ΩP) we define the Orlicz class119871Ω

(ΩP) as the set of all (equivalence classes of) measurablefunctions 119891 on Ω such that int

Ω

Ψ(|119891|119862)119889P lt infin for some0 lt 119862 lt infin We used to define the Morse-Transue space119872

Ψ

(ΩP) by

119872Ψ

(ΩP) = 119891Ω 997888rarr Cmeasurable

int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

119862

)119889P lt infin for any 119862 gt 0

(22)

4 Journal of Function Spaces

and we also introduce the following set

LΨ

(ΩP)

= 119891Ω 997888rarr Cmeasurable intΩ

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P lt infin

(23)

These three sets are not vector spaces and do not coincidebut we have

119872Ψ

(ΩP) subLΨ

(ΩP) sub 119871Ψ

(ΩP) (24)

We also define the Luxembourg gauge on 119871Ψ(ΩP) by

10038171003817100381710038171198911003817100381710038171003817Ψ= inf (120582 gt 0 int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

120582

) 119889P le 1) (25)

This functional is homogeneous and is 0 if and only if 119891 = 0but it is not subadditive

We say that the two functions Ψ1and Ψ

2as above are

equivalent if there exists some constant 119862 such that

119862Ψ1(119888119909) le Ψ

2(119909) le 119862

minus1

Ψ1(119862

minus1

119909) (26)

for any 119909 large enough Two equivalent functions define thesame Orlicz class with equivalent Luxembourg functionals

In order to define a good topology on 119871Ψ(ΩP) and to getproperties convenient for our purpose we will assume thatΨsatisfies the following definition

Definition 7 Let 0 lt 119901 le 1 One says that Ψ [0infin) rarr[0infin) is a growth function of order 119901 if it satisfies the twofollowing conditions

(1) Ψ is of lower type 119901 that is Ψ(119910119909) le 119910119901Ψ(119909) for any0 lt 119910 le 1 and at least for 119909 large enough

(2) 119909 997891rarr Ψ(119909)119909 is nonincreasing at least for every 119909large enough

We will say that Ψ is a growth function if it is a growthfunction of order 119901 for some 0 lt 119901 le 1

Sometimes we will use lower 119901 index for our growthfunction below

In particular a growth function Ψ is equivalent to thefunction 119909 997891rarr int119909

0

(Ψ(119904)119904)119889119904which is concave (see [12]) Nowfor such a concave growth function of order 119901

1198711

(ΩP) sub 119872Ψ

(ΩP) =LΨ

(ΩP) = 119871Ψ

sub 119871119901

(ΩP)

10038171003817100381710038171198911003817100381710038171003817Ψ≲ minint

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P (int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P)

119901

(27)

while

10038171003817100381710038171198911003817100381710038171003817Ψ= int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P ≲ max 100381710038171003817

10038171198911003817100381710038171003817Ψ10038171003817100381710038171198911003817100381710038171003817

119901

Ψ (28)

Moreover if we define 119889Ψ(119891 119892) = 119891 minus 119892

Ψ(119889Ψ(119891 119892) =

119891 minus 119892Ψ) then 119889

Ψand 119889Ψ (that we simply denote by sdot

Ψ

and sdot Ψ) are two equivalent metrics on 119871Ψ(ΩP) for whichit is completeWithout loss of generality we then assume thatevery growth function thatwe consider further is concave anddiffeomorphic

When we deal with spaces of holomorphic functionsit is very natural to require subharmonicity Then we willassume that Ψ is such that Ψ(|119891|) is subharmonic when119891 is holomorphic We will refer to such a function as asubharmonic-preserving function

Here are some examples of (concave) growth functionthat we may consider further

For Ψ a subharmonic-preserving growth function and120572 gt minus1 the weighted Bergman-Orlicz space 119860Ψ

120572of the ball

consists of those holomorphic functions onB119899which belongs

to the Orlicz space 119871Ψ(B119899 ]120572) where as usual for any real

number 120572 119889]120572(119911) = 119888

120572(1 minus |119911|

2

)120572

119889](119911) for |119911| lt 1 Here if120572 le minus1 119888

120572= 1 and if 120572 gt minus1 119888

120572= Γ(119899+1+120572)Γ(119899+1)Γ(120572+1)

is the normalizing constant so that ]120572has unit total mass

Further we will denote by sdot 120572Ψ

the correspond-ing Luxembourg (quasi)norm and by sdot Ψ

120572the quantity

intB119899

Ψ(|119891|)119889]120572 119860

Ψ

120572is metric space for the distance 119889

120572Ψor

119889Ψ

120572defined by respectively 119889

120572Ψ(119891 119892) = 119891 minus 119892

120572Ψand

119889Ψ

120572(119891 119892) = 119891 minus 119892

Ψ

120572 If Ψ(119905) = 119905119901 then we recover the

usual weighted Bergman space 119860119901120572 We have the following

inclusions

1198601

120572sub 119860

Ψ

120572sub 119860

119901

120572 (29)

whenever Ψ is a growth function of order 119901Note that the second inclusion which we see before leads

to another inclusion namely 119860Ψ120572 is embedded in 119860infin

(120572+119899+1)119901

This last inclusion will lead us to distance theorem Weshowed a complete analogue of a theorem in the unit diskbelow

It is important to mention that a linear operator 119879 from119860Ψ1

120572to 119883 with 119883 = 119871Ψ2(B

119899 ]120572) or 119883 = 119860Ψ2

120572 where Ψ

1and Ψ

2

are two growth functions is continuous if and only if it mapsa bounded set into a bounded set or equivalently if and onlyif there exists a constant 119862 gt 0 such that

max (1003817100381710038171003817119879 (119891)

1003817100381710038171003817120572Ψ2

1003817100381710038171003817119879 (119891)

1003817100381710038171003817

Ψ2

120572) le 119862 (30)

for any 119891 isin 119860Ψ1120572such that min(119891

120572Ψ1

119891Ψ1

120572) le 1 If119883 = C

then119879 is bounded if and only if |119879(119891)| le 119862 for119891 as previouslyThe next proposition says that the point evaluation

functionals are continuous on 119860Ψ120572

Proposition 8 (see [11]) Let 120572 gt minus1 and let Ψ be asubharmonic-preserving growth function For any 120572 isin B

119899and

any 119891 isin 119860Ψ120572 one has

1003816100381610038161003816119891 (119886)

1003816100381610038161003816le Ψ

minus1

((

2

1 minus |119886|

)

(119899+120572+1)

)10038171003817100381710038171198911003817100381710038171003817120572Ψ (31)

The proof is the same as that of [10 Proposition 19] andso is omitted (still use the hypothesis that Ψ(|119891|) is subhar-monic) We easily deduce from this and the completeness of119871Ψ the following result

Journal of Function Spaces 5

Corollary 9 (see [11]) 119860Ψ120572 endowed with 119889

120572Ψor 119889Ψ

120572 is a

complete metric space

Some facts from theory of 119899-harmonic function spaceswill be needed by us We will use the same notation for119872

119901(119891 119903) for harmonic and analytic functions

Definition 10 The quasinormed space 119871(119901 119902 120572) (0 lt 119901 119902 leinfin 120572 = (120572

1 120572

119899) 120572

119895gt minus1 119895 = 1 119899 is the set of those

functions 119891(119911) measurable in the polydisc U119899 for which thequasinorm

10038171003817100381710038171198911003817100381710038171003817119901119902120572

=

(int

119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895119902minus1

119872119902

119901(119891 119903)

119899

prod

119895=1

119889119903119895)

1119902

0 lt 119902 lt infin

ess sup119903isin119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895

119872119901(119891 119903)

119902 = infin

(32)

is finite For the subspace of 119871(119901 119902 120572) consisting of 119899-harmonic functions let ℎ(119901 119902 120572) = ℎ(U119899

) cap 119871(119901 119902 120572) (see[13])

For 119901 = 119902 lt infin the spaces ℎ(119901 119902 120572) coincide with thewell-known weighted Bergman spaces while for 119902 = infin theyare known as growth spaces

Let now 1 le 119901 119902 le infin120572119895gt 0 The following inclusion

will be used by us (see [13])

ℎ (119901 119902 120572) sub ℎ (infininfin 120572) (33)

This inclusion as in unit disk case (see [2]) allows us topose a distance problem in spaces of 119899-harmonic functionswe consider in this paper (we also did the same in spaces ofanalytic functions in the unit polydisk (see [1]))

Now we define Poisson kernel 119875120572in the unit disk

119875120572(119911) = Γ (120572 + 1) [Re 2

(1 minus 119911)120572+1minus 1] 119911 isin U 120572 ge 0

(34)

Let 119875120572(119911 120585) = 119875

120572(119911120585) For polydisk we have

119875120572(119911 120585) =

119899

prod

119895=1

119875120572119895

(119911119895 120585119895) (35)

where 120572 = (1205721 120572

119899) 120572

119895ge 0 120585 isin U119899

119911 isin U119899 119875120572is

119899-harmonic by both variables 119911 and 120585 and obviously holds119875120572(119911 120585) = 119875

120572(120585 119911) = 119875

120572(119911 120585) (see [13])

Reproducing integral formula Poisson-Bergman type willbe presented in the following theorem

Theorem B (see [13]) Let 120572119895gt 0 119906 isin ℎ(119901 119902 120572) If 0 lt 119901

119902 le infin 120573119895gt max120572

119895+ 1119901 minus 1 120572

119895 or 1 le 119901 le infin 0 lt 119902 le 1

120573119895ge 120572

119895(1 le 119895 le 119899)

119906 (119911) =

1

prod119899

119895=1Γ (120573

119895)

timesint

U119899

119899

prod

119895=1

(1 minus10038161003816100381610038161205771003816100381610038161003816

2

)

120573119895minus1

119875120573(119911 120577) 119906 (120577) 119889119898

2119899(120577)

119911 isin U119899

(36)

Let 119878120573120582

be a linear operator Bergman type

119878120573120582(119891) (119911)

=

(1 minus |119911|2

)

120582

Γ (120573 + 120582)

times int

U119899(1 minus10038161003816100381610038161205851003816100381610038161003816

2

)

120573minus1 10038161003816100381610038161003816119875120573+120582(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(37)

Theorem C (see [13]) Let 1 le 119901 119902 le infin 120572119895 120573

119895 120582

119895isin

R 120573119895gt 120572

119895gt minus120582

119895 1 le 119895 le 119899 Then the operator 119878

120573120582

continuously maps the space 119871(119901 119902 120572) into itself Moreoverthen operator 119878

120573120582is bounded in 119871(119901 119902 120572) if and only if 120573

119895gt

120572119895gt minus120582

119895 1 le 119895 le 119899

We will also need basic facts from theory of Bergmanspaces in minimal ball (see [14 15]) For the convenience ofreaders we will give some details about them

We consider the domain Blowastin C119899

119899 ge 2 defined by

Blowast= 119911 isin C

119899 |119911|2 + |119911 ∙ 119911| lt 1 (38)

where

119911 ∙ 119908 =

119899

sum

119895=1

119911119895119908119895 (39)

for 119911 and 119908 in C119899 This is the unit ball of C119899 with respect tothe norm

119873lowast(119911) = radic|119911|

2

+ |119911 ∙ 119911| 119911 isin C119899

(40)

The norm119873 = 119873lowastradic2was introduced by Hahn and Pflug in

[16] where it was shown to be the smallest norm in C119899 thatextends the Euclidean norm inR119899 under certain restrictions

The domainBlowasthas since been studied by Hahn and Pflug

[16] Mengotti and Youssfi [15] and Oeljeklaus and Youssfi[17] In particular it is well known that the automorphismgroup of B

lowastis compact and its identity component is

Aut0O (Blowast) = 119878

1

sdot 119878119874 (119899R) (41)

where the 1198781-action is diagonal and the 119878119874(119899R)-action isby matrix multiplication (see [17] eg) It is also well knownthat B

lowastis a nonhomogeneous domain Its singular boundary

consists of all points 119911 with 119911 ∙ 119911 = 0 and the regular part ofthe boundary of B

lowastconsists of strictly pseudoconvex points

6 Journal of Function Spaces

For each 119904 gt minus1 we let V119904denote the measure on B

lowast

defined by

119889V119904(119911) = (1 minus 119873

2

lowast(119911))

119904

119889V (119911) (42)

where V denotes the normalized Lebesguemeasure onBlowast For

all 0 lt 119901 lt infin we define

A119901

119904(B

lowast) =H (B

lowast) cap 119871

119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) (43)

whereH(Blowast) is the space of all holomorphic functions onB

lowast

Naturally the spaces A119901

119904(B

lowast) are called weighted Bergman

spaces of Blowast

When 119901 = 2 there exists an orthogonal projection from1198712

(Blowast 119889V

119904) onto A2

119904(B

lowast) This will be called the weighted

Bergman projection and is denoted by P119904Blowast

It is well known that P

119904Blowast

is an integral operator on1198712

(Blowast 119889V

119904) namely

P119904Blowast

119891 (119911) = int

Blowast

K119904Blowast

(119911 119908) 119891 (119908) 119889V119904(119908) (44)

where

K119904Blowast

(119911 119908) =

A (119883 119884)(119899

2+ 119899 minus 119904) V

119904(B

lowast) (119883

2minus 119884)

119899+1+119904(45)

is the weighted Bergman kernel Here

119883 = 1 minus 119911 ∙ 119908 119884 = (119911 ∙ 119911)119908 ∙ 119908 (46)

and A(119883 119884) is the suminfin

sum

119895=0

119888119899119904119895119883119899+119904minus1minus2119895

119884119895

times [2 (119899 + 119904)119883 minus

(119899 minus 2119895 + 119904) (119899 + 1 + 2119904)

119899 + 119904 + 1

(1198832

minus 119884)]

(47)

As usual

119888119899119904119895= (

119899 + 119904 + 1

2119895 + 1)

=

(119899 + 119904 + 1) (119899 + 119904) sdot sdot sdot (119899 + 119904 minus 2119895 + 1)

(2119895 + 1)

(48)

is the binomial coefficient and V119904(B

lowast) is the weighted

Lebesgue volume of Blowast See [15 17] for these formulas and

more information about the weighted Bergman kernelsBecause of the infinite sum 119860(119883 119884) the formula for K

119904Blowast

is not really a closed form As such it is inconvenient for usto do estimates for P

119904Blowast

directlyThus we employ a techniquethat was used in [15] More specifically we relate the domainBlowastto the hypersurface M of the Euclidean unit ball in C119899+1

defined by

M = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 |119911| lt 1 (49)

If Pr C119899+1

rarr C119899 is defined by

Pr (1199111 119911

119899 119911

119899+1) = (119911

1 119911

119899) (50)

and F = Pr|M then F M rarr B

lowastminus 0 Let

H = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 (51)

It was proved in [17] that there is an 119878119874(119899 + 1C)-invariantholomorphic form 120572 on H Moreover this form is uniqueup to a multiplicative constant In fact after appropriatenormalization the restriction to H cap (C 0)119899+1 of this formis given by

120572 (119911) =

119899+1

sum

119895=1

(minus1)119895minus1

119911119895

1198891199111and sdot sdot sdot and

119889119911

119895and sdot sdot sdot and 119889119911

119899+1 (52)

Our norm estimates for the weighted Bergman projectionP119904Blowast

on 119871119901 spaces will be based on the correspondingestimates onM Therefore we will consider the spaces 119871119901

119904(M)

consisting of measurable complex-valued functions 119891 on M

such that

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119904(M)= int

M

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

(1 minus |119911|2

)

119904120572 (119911) and 120572 (119911)

119862

lt infin (53)

where

119862 = (minus1)119899(119899+1)2

(2119894)119899

(54)

We use 119860119901119904(M) to denote the subspace of all holomorphic

functions in 119871119901119904(M) The weighted Bergman projection P

119904M

is then the orthogonal projection from 1198712119904(M) onto 1198602

119904(M)

Again it is well known that P119904M is an integral operator on

1198712

119904(M) given by the formula

P119904M119891 (119911) = int

M

K119904M (119911 119908) 119891 (119908) (1 minus |119908|

2

)

119904120572 (119911) and 120572 (119911)

119862

(55)

where K119904M is the corresponding Bergman kernel

The starting point for our analysis is the following closedform of K

119904M which was obtained as Theorem 32 in [15]

Lemma 11 (see [14]) The weighted Bergman kernel K119904M of

1198602

119904(M) is given by

K119904M (119911 119908) =

119862 (119899 minus 1 + (119899 + 1 + 2119904) 119911 ∙ 119908)

(1 minus 119911 ∙ 119908)119899+119904+1

(56)

where 119862 is a certain constant that depends on 119899 and 119904

Let 119891Blowastrarr C be a measurable function We define a

function T119891 onM by

(T119891) (119911) =119911119899+1

(2(119899 + 1)2

)

1119901

(119891 ∘ F) (119911)

=

119911119899+1119891 (119911

1 119911

119899)

(2(119899 + 1)2

)

1119901

(57)

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

For 119899 gt 1 the theorem was proved in [7] (when 120572 = 0)and in [8 9] (when 120572 gt minus1)

Inmonographs [5] one can find numerous applications ofthe formulas (10) and (11) in the complex analysis In [6] theauthor generalized Theorem A to the space with anisotropicweight function such as assuming that 1 le 119901 lt +infin 120572 =(120572

1 120572

119899) isin R119899 satisfies the conditions

120572119899gt minus1

120572119899+ 120572

119899minus1gt minus2

120572119899+ 120572

119899minus1+ 120572

119899minus2gt minus3

120572119899+ 120572

119899minus1+ 120572

119899minus2+ sdot sdot sdot + 120572

2gt minus (119899 minus 1)

120572119899+ 120572

119899minus1+ 120572

119899minus2+ sdot sdot sdot + 120572

2+ 120572

1gt minus119899

(12)

and then introduced the spaces 119860119901120572(B

119899) of functions 119891

holomorphic in B119899satisfying the condition

int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

lt +infin

(13)

For these anisotropic spaces similarities of the inte-gral representations (10) and (11) are obtained but in thementioned generalization special kernels 119878

12057311205732sdotsdotsdot120573119899

(119911 119908) equiv

119878120573(119911 119908) (where 120573

119894 1 le 119894 le 119899 are associated with 120572

119894 1 le

119894 le 119899 and 119901 in a special way) appear instead of 119888119899(120573) sdot (1 minus

⟨119911 119908⟩)minus(119899+1+120573) (see [6] Theorem 47)

Suppose now that 120572 = (1205721 120572

2 120572

119899) isin R119899 We put

119898120572= sum

120572119894lt0

minus 120572119894 119897

120572= sum

120572119894lt0

120572119894 (14)

We will write 120572 ≺ (lowast) only if the following conditions aresatisfied

120572119899+ 120572

119899minus1+ sdot sdot sdot + 120572

119894gt minus (119899 + 1 minus 119894) (1 le 119894 le 119899) (15)

Similarly if 120573 = (1205731 120573

2 120573

119899) isin C119899 then we will write 120573 ≺

(lowast) if only Re120573 equiv (Re1205731Re120573

2 Re120573

119899) ≺ (lowast)

We have the following lemmas

Lemma 2 (see [6]) If 120572 ≺ (lowast) then

int

B119899

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

=

120587119899

prod119899

119894=1(120572

119899+ 120572

119899minus1+ sdot sdot sdot + 120572

119894+ 119899 + 1 minus 119894)

lt +infin

(16)

We need also heavily the well-known Forelli-Rudin typeestimates for unit ball (see [5])

Proposition 3 Suppose 119888 gt 0 is real and 119905 gt minus1 Then theintegral

119869119888119905(119911) = int

B119899

(1 minus |119908|2

)

119905

119889] (119908)

|1 minus ⟨119911 119908⟩|119899+1+119905+119888

119911 isin B119899

(17)

has the following asymptotic property

119869119888119905sim (1 minus |119911|

2

)

minus119888

as |119911| 997888rarr 1 minus (18)

Definition 4 Assume that 0 lt 119901 lt +infin 120572 = (1205721 120572

119899) ≺

(lowast) Denote by 119871119901120572(B

119899) the space of all complex-valued

functions 119891 in B119899with

119901

120572(119891) equiv int

B119899

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

119901

times

119899

prod

119894=1

(1 minus10038161003816100381610038161199081

1003816100381610038161003816

2

minus10038161003816100381610038161199082

1003816100381610038161003816

2

minus sdot sdot sdot minus1003816100381610038161003816119908119894

1003816100381610038161003816

2

)

120572119894

119889] (119908)

lt infin

(19)

We obviously have

119901

120572(119891 + 119892) le 2

119901

(119901

120572(119891) +

119901

120572(119892))

119901

120572(119888119891) = 119888

119901

119901

120572(119891)

(20)

Lemma 5 (see [6]) Assume that 0 lt 119901 lt +infin 120572 ≺ (lowast) and119891 isin 119867(B

119899) Then

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

le

22119899+119897120572

119888119899sdot (1 minus |119911|)

2119899+119897120572

sdot 119901

120572(119891) forall119911 isin B

119899 (21)

where 119888119899= ](B

119899) is the volume of the unit ball of C119899

Note that this lemma leads to an embedding theoremimmediately which allows us to pose a distance problem asunit disk case

Corollary 6 (see [6]) 119867119901

120572(B

119899) is a closed subspace in 119871119901

120572(B

119899)

We will need some facts from theory of Bergman-Orliczspaces (see [10 11])

Let now Ψ [0infin) rarr [0infin) be a continuous non-decreasing function which vanishes and is continuous at 0Given a probabilistic space (ΩP) we define the Orlicz class119871Ω

(ΩP) as the set of all (equivalence classes of) measurablefunctions 119891 on Ω such that int

Ω

Ψ(|119891|119862)119889P lt infin for some0 lt 119862 lt infin We used to define the Morse-Transue space119872

Ψ

(ΩP) by

119872Ψ

(ΩP) = 119891Ω 997888rarr Cmeasurable

int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

119862

)119889P lt infin for any 119862 gt 0

(22)

4 Journal of Function Spaces

and we also introduce the following set

LΨ

(ΩP)

= 119891Ω 997888rarr Cmeasurable intΩ

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P lt infin

(23)

These three sets are not vector spaces and do not coincidebut we have

119872Ψ

(ΩP) subLΨ

(ΩP) sub 119871Ψ

(ΩP) (24)

We also define the Luxembourg gauge on 119871Ψ(ΩP) by

10038171003817100381710038171198911003817100381710038171003817Ψ= inf (120582 gt 0 int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

120582

) 119889P le 1) (25)

This functional is homogeneous and is 0 if and only if 119891 = 0but it is not subadditive

We say that the two functions Ψ1and Ψ

2as above are

equivalent if there exists some constant 119862 such that

119862Ψ1(119888119909) le Ψ

2(119909) le 119862

minus1

Ψ1(119862

minus1

119909) (26)

for any 119909 large enough Two equivalent functions define thesame Orlicz class with equivalent Luxembourg functionals

In order to define a good topology on 119871Ψ(ΩP) and to getproperties convenient for our purpose we will assume thatΨsatisfies the following definition

Definition 7 Let 0 lt 119901 le 1 One says that Ψ [0infin) rarr[0infin) is a growth function of order 119901 if it satisfies the twofollowing conditions

(1) Ψ is of lower type 119901 that is Ψ(119910119909) le 119910119901Ψ(119909) for any0 lt 119910 le 1 and at least for 119909 large enough

(2) 119909 997891rarr Ψ(119909)119909 is nonincreasing at least for every 119909large enough

We will say that Ψ is a growth function if it is a growthfunction of order 119901 for some 0 lt 119901 le 1

Sometimes we will use lower 119901 index for our growthfunction below

In particular a growth function Ψ is equivalent to thefunction 119909 997891rarr int119909

0

(Ψ(119904)119904)119889119904which is concave (see [12]) Nowfor such a concave growth function of order 119901

1198711

(ΩP) sub 119872Ψ

(ΩP) =LΨ

(ΩP) = 119871Ψ

sub 119871119901

(ΩP)

10038171003817100381710038171198911003817100381710038171003817Ψ≲ minint

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P (int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P)

119901

(27)

while

10038171003817100381710038171198911003817100381710038171003817Ψ= int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P ≲ max 100381710038171003817

10038171198911003817100381710038171003817Ψ10038171003817100381710038171198911003817100381710038171003817

119901

Ψ (28)

Moreover if we define 119889Ψ(119891 119892) = 119891 minus 119892

Ψ(119889Ψ(119891 119892) =

119891 minus 119892Ψ) then 119889

Ψand 119889Ψ (that we simply denote by sdot

Ψ

and sdot Ψ) are two equivalent metrics on 119871Ψ(ΩP) for whichit is completeWithout loss of generality we then assume thatevery growth function thatwe consider further is concave anddiffeomorphic

When we deal with spaces of holomorphic functionsit is very natural to require subharmonicity Then we willassume that Ψ is such that Ψ(|119891|) is subharmonic when119891 is holomorphic We will refer to such a function as asubharmonic-preserving function

Here are some examples of (concave) growth functionthat we may consider further

For Ψ a subharmonic-preserving growth function and120572 gt minus1 the weighted Bergman-Orlicz space 119860Ψ

120572of the ball

consists of those holomorphic functions onB119899which belongs

to the Orlicz space 119871Ψ(B119899 ]120572) where as usual for any real

number 120572 119889]120572(119911) = 119888

120572(1 minus |119911|

2

)120572

119889](119911) for |119911| lt 1 Here if120572 le minus1 119888

120572= 1 and if 120572 gt minus1 119888

120572= Γ(119899+1+120572)Γ(119899+1)Γ(120572+1)

is the normalizing constant so that ]120572has unit total mass

Further we will denote by sdot 120572Ψ

the correspond-ing Luxembourg (quasi)norm and by sdot Ψ

120572the quantity

intB119899

Ψ(|119891|)119889]120572 119860

Ψ

120572is metric space for the distance 119889

120572Ψor

119889Ψ

120572defined by respectively 119889

120572Ψ(119891 119892) = 119891 minus 119892

120572Ψand

119889Ψ

120572(119891 119892) = 119891 minus 119892

Ψ

120572 If Ψ(119905) = 119905119901 then we recover the

usual weighted Bergman space 119860119901120572 We have the following

inclusions

1198601

120572sub 119860

Ψ

120572sub 119860

119901

120572 (29)

whenever Ψ is a growth function of order 119901Note that the second inclusion which we see before leads

to another inclusion namely 119860Ψ120572 is embedded in 119860infin

(120572+119899+1)119901

This last inclusion will lead us to distance theorem Weshowed a complete analogue of a theorem in the unit diskbelow

It is important to mention that a linear operator 119879 from119860Ψ1

120572to 119883 with 119883 = 119871Ψ2(B

119899 ]120572) or 119883 = 119860Ψ2

120572 where Ψ

1and Ψ

2

are two growth functions is continuous if and only if it mapsa bounded set into a bounded set or equivalently if and onlyif there exists a constant 119862 gt 0 such that

max (1003817100381710038171003817119879 (119891)

1003817100381710038171003817120572Ψ2

1003817100381710038171003817119879 (119891)

1003817100381710038171003817

Ψ2

120572) le 119862 (30)

for any 119891 isin 119860Ψ1120572such that min(119891

120572Ψ1

119891Ψ1

120572) le 1 If119883 = C

then119879 is bounded if and only if |119879(119891)| le 119862 for119891 as previouslyThe next proposition says that the point evaluation

functionals are continuous on 119860Ψ120572

Proposition 8 (see [11]) Let 120572 gt minus1 and let Ψ be asubharmonic-preserving growth function For any 120572 isin B

119899and

any 119891 isin 119860Ψ120572 one has

1003816100381610038161003816119891 (119886)

1003816100381610038161003816le Ψ

minus1

((

2

1 minus |119886|

)

(119899+120572+1)

)10038171003817100381710038171198911003817100381710038171003817120572Ψ (31)

The proof is the same as that of [10 Proposition 19] andso is omitted (still use the hypothesis that Ψ(|119891|) is subhar-monic) We easily deduce from this and the completeness of119871Ψ the following result

Journal of Function Spaces 5

Corollary 9 (see [11]) 119860Ψ120572 endowed with 119889

120572Ψor 119889Ψ

120572 is a

complete metric space

Some facts from theory of 119899-harmonic function spaceswill be needed by us We will use the same notation for119872

119901(119891 119903) for harmonic and analytic functions

Definition 10 The quasinormed space 119871(119901 119902 120572) (0 lt 119901 119902 leinfin 120572 = (120572

1 120572

119899) 120572

119895gt minus1 119895 = 1 119899 is the set of those

functions 119891(119911) measurable in the polydisc U119899 for which thequasinorm

10038171003817100381710038171198911003817100381710038171003817119901119902120572

=

(int

119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895119902minus1

119872119902

119901(119891 119903)

119899

prod

119895=1

119889119903119895)

1119902

0 lt 119902 lt infin

ess sup119903isin119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895

119872119901(119891 119903)

119902 = infin

(32)

is finite For the subspace of 119871(119901 119902 120572) consisting of 119899-harmonic functions let ℎ(119901 119902 120572) = ℎ(U119899

) cap 119871(119901 119902 120572) (see[13])

For 119901 = 119902 lt infin the spaces ℎ(119901 119902 120572) coincide with thewell-known weighted Bergman spaces while for 119902 = infin theyare known as growth spaces

Let now 1 le 119901 119902 le infin120572119895gt 0 The following inclusion

will be used by us (see [13])

ℎ (119901 119902 120572) sub ℎ (infininfin 120572) (33)

This inclusion as in unit disk case (see [2]) allows us topose a distance problem in spaces of 119899-harmonic functionswe consider in this paper (we also did the same in spaces ofanalytic functions in the unit polydisk (see [1]))

Now we define Poisson kernel 119875120572in the unit disk

119875120572(119911) = Γ (120572 + 1) [Re 2

(1 minus 119911)120572+1minus 1] 119911 isin U 120572 ge 0

(34)

Let 119875120572(119911 120585) = 119875

120572(119911120585) For polydisk we have

119875120572(119911 120585) =

119899

prod

119895=1

119875120572119895

(119911119895 120585119895) (35)

where 120572 = (1205721 120572

119899) 120572

119895ge 0 120585 isin U119899

119911 isin U119899 119875120572is

119899-harmonic by both variables 119911 and 120585 and obviously holds119875120572(119911 120585) = 119875

120572(120585 119911) = 119875

120572(119911 120585) (see [13])

Reproducing integral formula Poisson-Bergman type willbe presented in the following theorem

Theorem B (see [13]) Let 120572119895gt 0 119906 isin ℎ(119901 119902 120572) If 0 lt 119901

119902 le infin 120573119895gt max120572

119895+ 1119901 minus 1 120572

119895 or 1 le 119901 le infin 0 lt 119902 le 1

120573119895ge 120572

119895(1 le 119895 le 119899)

119906 (119911) =

1

prod119899

119895=1Γ (120573

119895)

timesint

U119899

119899

prod

119895=1

(1 minus10038161003816100381610038161205771003816100381610038161003816

2

)

120573119895minus1

119875120573(119911 120577) 119906 (120577) 119889119898

2119899(120577)

119911 isin U119899

(36)

Let 119878120573120582

be a linear operator Bergman type

119878120573120582(119891) (119911)

=

(1 minus |119911|2

)

120582

Γ (120573 + 120582)

times int

U119899(1 minus10038161003816100381610038161205851003816100381610038161003816

2

)

120573minus1 10038161003816100381610038161003816119875120573+120582(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(37)

Theorem C (see [13]) Let 1 le 119901 119902 le infin 120572119895 120573

119895 120582

119895isin

R 120573119895gt 120572

119895gt minus120582

119895 1 le 119895 le 119899 Then the operator 119878

120573120582

continuously maps the space 119871(119901 119902 120572) into itself Moreoverthen operator 119878

120573120582is bounded in 119871(119901 119902 120572) if and only if 120573

119895gt

120572119895gt minus120582

119895 1 le 119895 le 119899

We will also need basic facts from theory of Bergmanspaces in minimal ball (see [14 15]) For the convenience ofreaders we will give some details about them

We consider the domain Blowastin C119899

119899 ge 2 defined by

Blowast= 119911 isin C

119899 |119911|2 + |119911 ∙ 119911| lt 1 (38)

where

119911 ∙ 119908 =

119899

sum

119895=1

119911119895119908119895 (39)

for 119911 and 119908 in C119899 This is the unit ball of C119899 with respect tothe norm

119873lowast(119911) = radic|119911|

2

+ |119911 ∙ 119911| 119911 isin C119899

(40)

The norm119873 = 119873lowastradic2was introduced by Hahn and Pflug in

[16] where it was shown to be the smallest norm in C119899 thatextends the Euclidean norm inR119899 under certain restrictions

The domainBlowasthas since been studied by Hahn and Pflug

[16] Mengotti and Youssfi [15] and Oeljeklaus and Youssfi[17] In particular it is well known that the automorphismgroup of B

lowastis compact and its identity component is

Aut0O (Blowast) = 119878

1

sdot 119878119874 (119899R) (41)

where the 1198781-action is diagonal and the 119878119874(119899R)-action isby matrix multiplication (see [17] eg) It is also well knownthat B

lowastis a nonhomogeneous domain Its singular boundary

consists of all points 119911 with 119911 ∙ 119911 = 0 and the regular part ofthe boundary of B

lowastconsists of strictly pseudoconvex points

6 Journal of Function Spaces

For each 119904 gt minus1 we let V119904denote the measure on B

lowast

defined by

119889V119904(119911) = (1 minus 119873

2

lowast(119911))

119904

119889V (119911) (42)

where V denotes the normalized Lebesguemeasure onBlowast For

all 0 lt 119901 lt infin we define

A119901

119904(B

lowast) =H (B

lowast) cap 119871

119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) (43)

whereH(Blowast) is the space of all holomorphic functions onB

lowast

Naturally the spaces A119901

119904(B

lowast) are called weighted Bergman

spaces of Blowast

When 119901 = 2 there exists an orthogonal projection from1198712

(Blowast 119889V

119904) onto A2

119904(B

lowast) This will be called the weighted

Bergman projection and is denoted by P119904Blowast

It is well known that P

119904Blowast

is an integral operator on1198712

(Blowast 119889V

119904) namely

P119904Blowast

119891 (119911) = int

Blowast

K119904Blowast

(119911 119908) 119891 (119908) 119889V119904(119908) (44)

where

K119904Blowast

(119911 119908) =

A (119883 119884)(119899

2+ 119899 minus 119904) V

119904(B

lowast) (119883

2minus 119884)

119899+1+119904(45)

is the weighted Bergman kernel Here

119883 = 1 minus 119911 ∙ 119908 119884 = (119911 ∙ 119911)119908 ∙ 119908 (46)

and A(119883 119884) is the suminfin

sum

119895=0

119888119899119904119895119883119899+119904minus1minus2119895

119884119895

times [2 (119899 + 119904)119883 minus

(119899 minus 2119895 + 119904) (119899 + 1 + 2119904)

119899 + 119904 + 1

(1198832

minus 119884)]

(47)

As usual

119888119899119904119895= (

119899 + 119904 + 1

2119895 + 1)

=

(119899 + 119904 + 1) (119899 + 119904) sdot sdot sdot (119899 + 119904 minus 2119895 + 1)

(2119895 + 1)

(48)

is the binomial coefficient and V119904(B

lowast) is the weighted

Lebesgue volume of Blowast See [15 17] for these formulas and

more information about the weighted Bergman kernelsBecause of the infinite sum 119860(119883 119884) the formula for K

119904Blowast

is not really a closed form As such it is inconvenient for usto do estimates for P

119904Blowast

directlyThus we employ a techniquethat was used in [15] More specifically we relate the domainBlowastto the hypersurface M of the Euclidean unit ball in C119899+1

defined by

M = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 |119911| lt 1 (49)

If Pr C119899+1

rarr C119899 is defined by

Pr (1199111 119911

119899 119911

119899+1) = (119911

1 119911

119899) (50)

and F = Pr|M then F M rarr B

lowastminus 0 Let

H = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 (51)

It was proved in [17] that there is an 119878119874(119899 + 1C)-invariantholomorphic form 120572 on H Moreover this form is uniqueup to a multiplicative constant In fact after appropriatenormalization the restriction to H cap (C 0)119899+1 of this formis given by

120572 (119911) =

119899+1

sum

119895=1

(minus1)119895minus1

119911119895

1198891199111and sdot sdot sdot and

119889119911

119895and sdot sdot sdot and 119889119911

119899+1 (52)

Our norm estimates for the weighted Bergman projectionP119904Blowast

on 119871119901 spaces will be based on the correspondingestimates onM Therefore we will consider the spaces 119871119901

119904(M)

consisting of measurable complex-valued functions 119891 on M

such that

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119904(M)= int

M

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

(1 minus |119911|2

)

119904120572 (119911) and 120572 (119911)

119862

lt infin (53)

where

119862 = (minus1)119899(119899+1)2

(2119894)119899

(54)

We use 119860119901119904(M) to denote the subspace of all holomorphic

functions in 119871119901119904(M) The weighted Bergman projection P

119904M

is then the orthogonal projection from 1198712119904(M) onto 1198602

119904(M)

Again it is well known that P119904M is an integral operator on

1198712

119904(M) given by the formula

P119904M119891 (119911) = int

M

K119904M (119911 119908) 119891 (119908) (1 minus |119908|

2

)

119904120572 (119911) and 120572 (119911)

119862

(55)

where K119904M is the corresponding Bergman kernel

The starting point for our analysis is the following closedform of K

119904M which was obtained as Theorem 32 in [15]

Lemma 11 (see [14]) The weighted Bergman kernel K119904M of

1198602

119904(M) is given by

K119904M (119911 119908) =

119862 (119899 minus 1 + (119899 + 1 + 2119904) 119911 ∙ 119908)

(1 minus 119911 ∙ 119908)119899+119904+1

(56)

where 119862 is a certain constant that depends on 119899 and 119904

Let 119891Blowastrarr C be a measurable function We define a

function T119891 onM by

(T119891) (119911) =119911119899+1

(2(119899 + 1)2

)

1119901

(119891 ∘ F) (119911)

=

119911119899+1119891 (119911

1 119911

119899)

(2(119899 + 1)2

)

1119901

(57)

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

and we also introduce the following set

LΨ

(ΩP)

= 119891Ω 997888rarr Cmeasurable intΩ

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P lt infin

(23)

These three sets are not vector spaces and do not coincidebut we have

119872Ψ

(ΩP) subLΨ

(ΩP) sub 119871Ψ

(ΩP) (24)

We also define the Luxembourg gauge on 119871Ψ(ΩP) by

10038171003817100381710038171198911003817100381710038171003817Ψ= inf (120582 gt 0 int

Ω

Ψ(

10038161003816100381610038161198911003816100381610038161003816

120582

) 119889P le 1) (25)

This functional is homogeneous and is 0 if and only if 119891 = 0but it is not subadditive

We say that the two functions Ψ1and Ψ

2as above are

equivalent if there exists some constant 119862 such that

119862Ψ1(119888119909) le Ψ

2(119909) le 119862

minus1

Ψ1(119862

minus1

119909) (26)

for any 119909 large enough Two equivalent functions define thesame Orlicz class with equivalent Luxembourg functionals

In order to define a good topology on 119871Ψ(ΩP) and to getproperties convenient for our purpose we will assume thatΨsatisfies the following definition

Definition 7 Let 0 lt 119901 le 1 One says that Ψ [0infin) rarr[0infin) is a growth function of order 119901 if it satisfies the twofollowing conditions

(1) Ψ is of lower type 119901 that is Ψ(119910119909) le 119910119901Ψ(119909) for any0 lt 119910 le 1 and at least for 119909 large enough

(2) 119909 997891rarr Ψ(119909)119909 is nonincreasing at least for every 119909large enough

We will say that Ψ is a growth function if it is a growthfunction of order 119901 for some 0 lt 119901 le 1

Sometimes we will use lower 119901 index for our growthfunction below

In particular a growth function Ψ is equivalent to thefunction 119909 997891rarr int119909

0

(Ψ(119904)119904)119889119904which is concave (see [12]) Nowfor such a concave growth function of order 119901

1198711

(ΩP) sub 119872Ψ

(ΩP) =LΨ

(ΩP) = 119871Ψ

sub 119871119901

(ΩP)

10038171003817100381710038171198911003817100381710038171003817Ψ≲ minint

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P (int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P)

119901

(27)

while

10038171003817100381710038171198911003817100381710038171003817Ψ= int

Ω

Ψ (10038161003816100381610038161198911003816100381610038161003816) 119889P ≲ max 100381710038171003817

10038171198911003817100381710038171003817Ψ10038171003817100381710038171198911003817100381710038171003817

119901

Ψ (28)

Moreover if we define 119889Ψ(119891 119892) = 119891 minus 119892

Ψ(119889Ψ(119891 119892) =

119891 minus 119892Ψ) then 119889

Ψand 119889Ψ (that we simply denote by sdot

Ψ

and sdot Ψ) are two equivalent metrics on 119871Ψ(ΩP) for whichit is completeWithout loss of generality we then assume thatevery growth function thatwe consider further is concave anddiffeomorphic

When we deal with spaces of holomorphic functionsit is very natural to require subharmonicity Then we willassume that Ψ is such that Ψ(|119891|) is subharmonic when119891 is holomorphic We will refer to such a function as asubharmonic-preserving function

Here are some examples of (concave) growth functionthat we may consider further

For Ψ a subharmonic-preserving growth function and120572 gt minus1 the weighted Bergman-Orlicz space 119860Ψ

120572of the ball

consists of those holomorphic functions onB119899which belongs

to the Orlicz space 119871Ψ(B119899 ]120572) where as usual for any real

number 120572 119889]120572(119911) = 119888

120572(1 minus |119911|

2

)120572

119889](119911) for |119911| lt 1 Here if120572 le minus1 119888

120572= 1 and if 120572 gt minus1 119888

120572= Γ(119899+1+120572)Γ(119899+1)Γ(120572+1)

is the normalizing constant so that ]120572has unit total mass

Further we will denote by sdot 120572Ψ

the correspond-ing Luxembourg (quasi)norm and by sdot Ψ

120572the quantity

intB119899

Ψ(|119891|)119889]120572 119860

Ψ

120572is metric space for the distance 119889

120572Ψor

119889Ψ

120572defined by respectively 119889

120572Ψ(119891 119892) = 119891 minus 119892

120572Ψand

119889Ψ

120572(119891 119892) = 119891 minus 119892

Ψ

120572 If Ψ(119905) = 119905119901 then we recover the

usual weighted Bergman space 119860119901120572 We have the following

inclusions

1198601

120572sub 119860

Ψ

120572sub 119860

119901

120572 (29)

whenever Ψ is a growth function of order 119901Note that the second inclusion which we see before leads

to another inclusion namely 119860Ψ120572 is embedded in 119860infin

(120572+119899+1)119901

This last inclusion will lead us to distance theorem Weshowed a complete analogue of a theorem in the unit diskbelow

It is important to mention that a linear operator 119879 from119860Ψ1

120572to 119883 with 119883 = 119871Ψ2(B

119899 ]120572) or 119883 = 119860Ψ2

120572 where Ψ

1and Ψ

2

are two growth functions is continuous if and only if it mapsa bounded set into a bounded set or equivalently if and onlyif there exists a constant 119862 gt 0 such that

max (1003817100381710038171003817119879 (119891)

1003817100381710038171003817120572Ψ2

1003817100381710038171003817119879 (119891)

1003817100381710038171003817

Ψ2

120572) le 119862 (30)

for any 119891 isin 119860Ψ1120572such that min(119891

120572Ψ1

119891Ψ1

120572) le 1 If119883 = C

then119879 is bounded if and only if |119879(119891)| le 119862 for119891 as previouslyThe next proposition says that the point evaluation

functionals are continuous on 119860Ψ120572

Proposition 8 (see [11]) Let 120572 gt minus1 and let Ψ be asubharmonic-preserving growth function For any 120572 isin B

119899and

any 119891 isin 119860Ψ120572 one has

1003816100381610038161003816119891 (119886)

1003816100381610038161003816le Ψ

minus1

((

2

1 minus |119886|

)

(119899+120572+1)

)10038171003817100381710038171198911003817100381710038171003817120572Ψ (31)

The proof is the same as that of [10 Proposition 19] andso is omitted (still use the hypothesis that Ψ(|119891|) is subhar-monic) We easily deduce from this and the completeness of119871Ψ the following result

Journal of Function Spaces 5

Corollary 9 (see [11]) 119860Ψ120572 endowed with 119889

120572Ψor 119889Ψ

120572 is a

complete metric space

Some facts from theory of 119899-harmonic function spaceswill be needed by us We will use the same notation for119872

119901(119891 119903) for harmonic and analytic functions

Definition 10 The quasinormed space 119871(119901 119902 120572) (0 lt 119901 119902 leinfin 120572 = (120572

1 120572

119899) 120572

119895gt minus1 119895 = 1 119899 is the set of those

functions 119891(119911) measurable in the polydisc U119899 for which thequasinorm

10038171003817100381710038171198911003817100381710038171003817119901119902120572

=

(int

119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895119902minus1

119872119902

119901(119891 119903)

119899

prod

119895=1

119889119903119895)

1119902

0 lt 119902 lt infin

ess sup119903isin119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895

119872119901(119891 119903)

119902 = infin

(32)

is finite For the subspace of 119871(119901 119902 120572) consisting of 119899-harmonic functions let ℎ(119901 119902 120572) = ℎ(U119899

) cap 119871(119901 119902 120572) (see[13])

For 119901 = 119902 lt infin the spaces ℎ(119901 119902 120572) coincide with thewell-known weighted Bergman spaces while for 119902 = infin theyare known as growth spaces

Let now 1 le 119901 119902 le infin120572119895gt 0 The following inclusion

will be used by us (see [13])

ℎ (119901 119902 120572) sub ℎ (infininfin 120572) (33)

This inclusion as in unit disk case (see [2]) allows us topose a distance problem in spaces of 119899-harmonic functionswe consider in this paper (we also did the same in spaces ofanalytic functions in the unit polydisk (see [1]))

Now we define Poisson kernel 119875120572in the unit disk

119875120572(119911) = Γ (120572 + 1) [Re 2

(1 minus 119911)120572+1minus 1] 119911 isin U 120572 ge 0

(34)

Let 119875120572(119911 120585) = 119875

120572(119911120585) For polydisk we have

119875120572(119911 120585) =

119899

prod

119895=1

119875120572119895

(119911119895 120585119895) (35)

where 120572 = (1205721 120572

119899) 120572

119895ge 0 120585 isin U119899

119911 isin U119899 119875120572is

119899-harmonic by both variables 119911 and 120585 and obviously holds119875120572(119911 120585) = 119875

120572(120585 119911) = 119875

120572(119911 120585) (see [13])

Reproducing integral formula Poisson-Bergman type willbe presented in the following theorem

Theorem B (see [13]) Let 120572119895gt 0 119906 isin ℎ(119901 119902 120572) If 0 lt 119901

119902 le infin 120573119895gt max120572

119895+ 1119901 minus 1 120572

119895 or 1 le 119901 le infin 0 lt 119902 le 1

120573119895ge 120572

119895(1 le 119895 le 119899)

119906 (119911) =

1

prod119899

119895=1Γ (120573

119895)

timesint

U119899

119899

prod

119895=1

(1 minus10038161003816100381610038161205771003816100381610038161003816

2

)

120573119895minus1

119875120573(119911 120577) 119906 (120577) 119889119898

2119899(120577)

119911 isin U119899

(36)

Let 119878120573120582

be a linear operator Bergman type

119878120573120582(119891) (119911)

=

(1 minus |119911|2

)

120582

Γ (120573 + 120582)

times int

U119899(1 minus10038161003816100381610038161205851003816100381610038161003816

2

)

120573minus1 10038161003816100381610038161003816119875120573+120582(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(37)

Theorem C (see [13]) Let 1 le 119901 119902 le infin 120572119895 120573

119895 120582

119895isin

R 120573119895gt 120572

119895gt minus120582

119895 1 le 119895 le 119899 Then the operator 119878

120573120582

continuously maps the space 119871(119901 119902 120572) into itself Moreoverthen operator 119878

120573120582is bounded in 119871(119901 119902 120572) if and only if 120573

119895gt

120572119895gt minus120582

119895 1 le 119895 le 119899

We will also need basic facts from theory of Bergmanspaces in minimal ball (see [14 15]) For the convenience ofreaders we will give some details about them

We consider the domain Blowastin C119899

119899 ge 2 defined by

Blowast= 119911 isin C

119899 |119911|2 + |119911 ∙ 119911| lt 1 (38)

where

119911 ∙ 119908 =

119899

sum

119895=1

119911119895119908119895 (39)

for 119911 and 119908 in C119899 This is the unit ball of C119899 with respect tothe norm

119873lowast(119911) = radic|119911|

2

+ |119911 ∙ 119911| 119911 isin C119899

(40)

The norm119873 = 119873lowastradic2was introduced by Hahn and Pflug in

[16] where it was shown to be the smallest norm in C119899 thatextends the Euclidean norm inR119899 under certain restrictions

The domainBlowasthas since been studied by Hahn and Pflug

[16] Mengotti and Youssfi [15] and Oeljeklaus and Youssfi[17] In particular it is well known that the automorphismgroup of B

lowastis compact and its identity component is

Aut0O (Blowast) = 119878

1

sdot 119878119874 (119899R) (41)

where the 1198781-action is diagonal and the 119878119874(119899R)-action isby matrix multiplication (see [17] eg) It is also well knownthat B

lowastis a nonhomogeneous domain Its singular boundary

consists of all points 119911 with 119911 ∙ 119911 = 0 and the regular part ofthe boundary of B

lowastconsists of strictly pseudoconvex points

6 Journal of Function Spaces

For each 119904 gt minus1 we let V119904denote the measure on B

lowast

defined by

119889V119904(119911) = (1 minus 119873

2

lowast(119911))

119904

119889V (119911) (42)

where V denotes the normalized Lebesguemeasure onBlowast For

all 0 lt 119901 lt infin we define

A119901

119904(B

lowast) =H (B

lowast) cap 119871

119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) (43)

whereH(Blowast) is the space of all holomorphic functions onB

lowast

Naturally the spaces A119901

119904(B

lowast) are called weighted Bergman

spaces of Blowast

When 119901 = 2 there exists an orthogonal projection from1198712

(Blowast 119889V

119904) onto A2

119904(B

lowast) This will be called the weighted

Bergman projection and is denoted by P119904Blowast

It is well known that P

119904Blowast

is an integral operator on1198712

(Blowast 119889V

119904) namely

P119904Blowast

119891 (119911) = int

Blowast

K119904Blowast

(119911 119908) 119891 (119908) 119889V119904(119908) (44)

where

K119904Blowast

(119911 119908) =

A (119883 119884)(119899

2+ 119899 minus 119904) V

119904(B

lowast) (119883

2minus 119884)

119899+1+119904(45)

is the weighted Bergman kernel Here

119883 = 1 minus 119911 ∙ 119908 119884 = (119911 ∙ 119911)119908 ∙ 119908 (46)

and A(119883 119884) is the suminfin

sum

119895=0

119888119899119904119895119883119899+119904minus1minus2119895

119884119895

times [2 (119899 + 119904)119883 minus

(119899 minus 2119895 + 119904) (119899 + 1 + 2119904)

119899 + 119904 + 1

(1198832

minus 119884)]

(47)

As usual

119888119899119904119895= (

119899 + 119904 + 1

2119895 + 1)

=

(119899 + 119904 + 1) (119899 + 119904) sdot sdot sdot (119899 + 119904 minus 2119895 + 1)

(2119895 + 1)

(48)

is the binomial coefficient and V119904(B

lowast) is the weighted

Lebesgue volume of Blowast See [15 17] for these formulas and

more information about the weighted Bergman kernelsBecause of the infinite sum 119860(119883 119884) the formula for K

119904Blowast

is not really a closed form As such it is inconvenient for usto do estimates for P

119904Blowast

directlyThus we employ a techniquethat was used in [15] More specifically we relate the domainBlowastto the hypersurface M of the Euclidean unit ball in C119899+1

defined by

M = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 |119911| lt 1 (49)

If Pr C119899+1

rarr C119899 is defined by

Pr (1199111 119911

119899 119911

119899+1) = (119911

1 119911

119899) (50)

and F = Pr|M then F M rarr B

lowastminus 0 Let

H = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 (51)

It was proved in [17] that there is an 119878119874(119899 + 1C)-invariantholomorphic form 120572 on H Moreover this form is uniqueup to a multiplicative constant In fact after appropriatenormalization the restriction to H cap (C 0)119899+1 of this formis given by

120572 (119911) =

119899+1

sum

119895=1

(minus1)119895minus1

119911119895

1198891199111and sdot sdot sdot and

119889119911

119895and sdot sdot sdot and 119889119911

119899+1 (52)

Our norm estimates for the weighted Bergman projectionP119904Blowast

on 119871119901 spaces will be based on the correspondingestimates onM Therefore we will consider the spaces 119871119901

119904(M)

consisting of measurable complex-valued functions 119891 on M

such that

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119904(M)= int

M

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

(1 minus |119911|2

)

119904120572 (119911) and 120572 (119911)

119862

lt infin (53)

where

119862 = (minus1)119899(119899+1)2

(2119894)119899

(54)

We use 119860119901119904(M) to denote the subspace of all holomorphic

functions in 119871119901119904(M) The weighted Bergman projection P

119904M

is then the orthogonal projection from 1198712119904(M) onto 1198602

119904(M)

Again it is well known that P119904M is an integral operator on

1198712

119904(M) given by the formula

P119904M119891 (119911) = int

M

K119904M (119911 119908) 119891 (119908) (1 minus |119908|

2

)

119904120572 (119911) and 120572 (119911)

119862

(55)

where K119904M is the corresponding Bergman kernel

The starting point for our analysis is the following closedform of K

119904M which was obtained as Theorem 32 in [15]

Lemma 11 (see [14]) The weighted Bergman kernel K119904M of

1198602

119904(M) is given by

K119904M (119911 119908) =

119862 (119899 minus 1 + (119899 + 1 + 2119904) 119911 ∙ 119908)

(1 minus 119911 ∙ 119908)119899+119904+1

(56)

where 119862 is a certain constant that depends on 119899 and 119904

Let 119891Blowastrarr C be a measurable function We define a

function T119891 onM by

(T119891) (119911) =119911119899+1

(2(119899 + 1)2

)

1119901

(119891 ∘ F) (119911)

=

119911119899+1119891 (119911

1 119911

119899)

(2(119899 + 1)2

)

1119901

(57)

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

Corollary 9 (see [11]) 119860Ψ120572 endowed with 119889

120572Ψor 119889Ψ

120572 is a

complete metric space

Some facts from theory of 119899-harmonic function spaceswill be needed by us We will use the same notation for119872

119901(119891 119903) for harmonic and analytic functions

Definition 10 The quasinormed space 119871(119901 119902 120572) (0 lt 119901 119902 leinfin 120572 = (120572

1 120572

119899) 120572

119895gt minus1 119895 = 1 119899 is the set of those

functions 119891(119911) measurable in the polydisc U119899 for which thequasinorm

10038171003817100381710038171198911003817100381710038171003817119901119902120572

=

(int

119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895119902minus1

119872119902

119901(119891 119903)

119899

prod

119895=1

119889119903119895)

1119902

0 lt 119902 lt infin

ess sup119903isin119868119899

119899

prod

119895=1

(1 minus 119903119895)

120572119895

119872119901(119891 119903)

119902 = infin

(32)

is finite For the subspace of 119871(119901 119902 120572) consisting of 119899-harmonic functions let ℎ(119901 119902 120572) = ℎ(U119899

) cap 119871(119901 119902 120572) (see[13])

For 119901 = 119902 lt infin the spaces ℎ(119901 119902 120572) coincide with thewell-known weighted Bergman spaces while for 119902 = infin theyare known as growth spaces

Let now 1 le 119901 119902 le infin120572119895gt 0 The following inclusion

will be used by us (see [13])

ℎ (119901 119902 120572) sub ℎ (infininfin 120572) (33)

This inclusion as in unit disk case (see [2]) allows us topose a distance problem in spaces of 119899-harmonic functionswe consider in this paper (we also did the same in spaces ofanalytic functions in the unit polydisk (see [1]))

Now we define Poisson kernel 119875120572in the unit disk

119875120572(119911) = Γ (120572 + 1) [Re 2

(1 minus 119911)120572+1minus 1] 119911 isin U 120572 ge 0

(34)

Let 119875120572(119911 120585) = 119875

120572(119911120585) For polydisk we have

119875120572(119911 120585) =

119899

prod

119895=1

119875120572119895

(119911119895 120585119895) (35)

where 120572 = (1205721 120572

119899) 120572

119895ge 0 120585 isin U119899

119911 isin U119899 119875120572is

119899-harmonic by both variables 119911 and 120585 and obviously holds119875120572(119911 120585) = 119875

120572(120585 119911) = 119875

120572(119911 120585) (see [13])

Reproducing integral formula Poisson-Bergman type willbe presented in the following theorem

Theorem B (see [13]) Let 120572119895gt 0 119906 isin ℎ(119901 119902 120572) If 0 lt 119901

119902 le infin 120573119895gt max120572

119895+ 1119901 minus 1 120572

119895 or 1 le 119901 le infin 0 lt 119902 le 1

120573119895ge 120572

119895(1 le 119895 le 119899)

119906 (119911) =

1

prod119899

119895=1Γ (120573

119895)

timesint

U119899

119899

prod

119895=1

(1 minus10038161003816100381610038161205771003816100381610038161003816

2

)

120573119895minus1

119875120573(119911 120577) 119906 (120577) 119889119898

2119899(120577)

119911 isin U119899

(36)

Let 119878120573120582

be a linear operator Bergman type

119878120573120582(119891) (119911)

=

(1 minus |119911|2

)

120582

Γ (120573 + 120582)

times int

U119899(1 minus10038161003816100381610038161205851003816100381610038161003816

2

)

120573minus1 10038161003816100381610038161003816119875120573+120582(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(37)

Theorem C (see [13]) Let 1 le 119901 119902 le infin 120572119895 120573

119895 120582

119895isin

R 120573119895gt 120572

119895gt minus120582

119895 1 le 119895 le 119899 Then the operator 119878

120573120582

continuously maps the space 119871(119901 119902 120572) into itself Moreoverthen operator 119878

120573120582is bounded in 119871(119901 119902 120572) if and only if 120573

119895gt

120572119895gt minus120582

119895 1 le 119895 le 119899

We will also need basic facts from theory of Bergmanspaces in minimal ball (see [14 15]) For the convenience ofreaders we will give some details about them

We consider the domain Blowastin C119899

119899 ge 2 defined by

Blowast= 119911 isin C

119899 |119911|2 + |119911 ∙ 119911| lt 1 (38)

where

119911 ∙ 119908 =

119899

sum

119895=1

119911119895119908119895 (39)

for 119911 and 119908 in C119899 This is the unit ball of C119899 with respect tothe norm

119873lowast(119911) = radic|119911|

2

+ |119911 ∙ 119911| 119911 isin C119899

(40)

The norm119873 = 119873lowastradic2was introduced by Hahn and Pflug in

[16] where it was shown to be the smallest norm in C119899 thatextends the Euclidean norm inR119899 under certain restrictions

The domainBlowasthas since been studied by Hahn and Pflug

[16] Mengotti and Youssfi [15] and Oeljeklaus and Youssfi[17] In particular it is well known that the automorphismgroup of B

lowastis compact and its identity component is

Aut0O (Blowast) = 119878

1

sdot 119878119874 (119899R) (41)

where the 1198781-action is diagonal and the 119878119874(119899R)-action isby matrix multiplication (see [17] eg) It is also well knownthat B

lowastis a nonhomogeneous domain Its singular boundary

consists of all points 119911 with 119911 ∙ 119911 = 0 and the regular part ofthe boundary of B

lowastconsists of strictly pseudoconvex points

6 Journal of Function Spaces

For each 119904 gt minus1 we let V119904denote the measure on B

lowast

defined by

119889V119904(119911) = (1 minus 119873

2

lowast(119911))

119904

119889V (119911) (42)

where V denotes the normalized Lebesguemeasure onBlowast For

all 0 lt 119901 lt infin we define

A119901

119904(B

lowast) =H (B

lowast) cap 119871

119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) (43)

whereH(Blowast) is the space of all holomorphic functions onB

lowast

Naturally the spaces A119901

119904(B

lowast) are called weighted Bergman

spaces of Blowast

When 119901 = 2 there exists an orthogonal projection from1198712

(Blowast 119889V

119904) onto A2

119904(B

lowast) This will be called the weighted

Bergman projection and is denoted by P119904Blowast

It is well known that P

119904Blowast

is an integral operator on1198712

(Blowast 119889V

119904) namely

P119904Blowast

119891 (119911) = int

Blowast

K119904Blowast

(119911 119908) 119891 (119908) 119889V119904(119908) (44)

where

K119904Blowast

(119911 119908) =

A (119883 119884)(119899

2+ 119899 minus 119904) V

119904(B

lowast) (119883

2minus 119884)

119899+1+119904(45)

is the weighted Bergman kernel Here

119883 = 1 minus 119911 ∙ 119908 119884 = (119911 ∙ 119911)119908 ∙ 119908 (46)

and A(119883 119884) is the suminfin

sum

119895=0

119888119899119904119895119883119899+119904minus1minus2119895

119884119895

times [2 (119899 + 119904)119883 minus

(119899 minus 2119895 + 119904) (119899 + 1 + 2119904)

119899 + 119904 + 1

(1198832

minus 119884)]

(47)

As usual

119888119899119904119895= (

119899 + 119904 + 1

2119895 + 1)

=

(119899 + 119904 + 1) (119899 + 119904) sdot sdot sdot (119899 + 119904 minus 2119895 + 1)

(2119895 + 1)

(48)

is the binomial coefficient and V119904(B

lowast) is the weighted

Lebesgue volume of Blowast See [15 17] for these formulas and

more information about the weighted Bergman kernelsBecause of the infinite sum 119860(119883 119884) the formula for K

119904Blowast

is not really a closed form As such it is inconvenient for usto do estimates for P

119904Blowast

directlyThus we employ a techniquethat was used in [15] More specifically we relate the domainBlowastto the hypersurface M of the Euclidean unit ball in C119899+1

defined by

M = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 |119911| lt 1 (49)

If Pr C119899+1

rarr C119899 is defined by

Pr (1199111 119911

119899 119911

119899+1) = (119911

1 119911

119899) (50)

and F = Pr|M then F M rarr B

lowastminus 0 Let

H = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 (51)

It was proved in [17] that there is an 119878119874(119899 + 1C)-invariantholomorphic form 120572 on H Moreover this form is uniqueup to a multiplicative constant In fact after appropriatenormalization the restriction to H cap (C 0)119899+1 of this formis given by

120572 (119911) =

119899+1

sum

119895=1

(minus1)119895minus1

119911119895

1198891199111and sdot sdot sdot and

119889119911

119895and sdot sdot sdot and 119889119911

119899+1 (52)

Our norm estimates for the weighted Bergman projectionP119904Blowast

on 119871119901 spaces will be based on the correspondingestimates onM Therefore we will consider the spaces 119871119901

119904(M)

consisting of measurable complex-valued functions 119891 on M

such that

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119904(M)= int

M

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

(1 minus |119911|2

)

119904120572 (119911) and 120572 (119911)

119862

lt infin (53)

where

119862 = (minus1)119899(119899+1)2

(2119894)119899

(54)

We use 119860119901119904(M) to denote the subspace of all holomorphic

functions in 119871119901119904(M) The weighted Bergman projection P

119904M

is then the orthogonal projection from 1198712119904(M) onto 1198602

119904(M)

Again it is well known that P119904M is an integral operator on

1198712

119904(M) given by the formula

P119904M119891 (119911) = int

M

K119904M (119911 119908) 119891 (119908) (1 minus |119908|

2

)

119904120572 (119911) and 120572 (119911)

119862

(55)

where K119904M is the corresponding Bergman kernel

The starting point for our analysis is the following closedform of K

119904M which was obtained as Theorem 32 in [15]

Lemma 11 (see [14]) The weighted Bergman kernel K119904M of

1198602

119904(M) is given by

K119904M (119911 119908) =

119862 (119899 minus 1 + (119899 + 1 + 2119904) 119911 ∙ 119908)

(1 minus 119911 ∙ 119908)119899+119904+1

(56)

where 119862 is a certain constant that depends on 119899 and 119904

Let 119891Blowastrarr C be a measurable function We define a

function T119891 onM by

(T119891) (119911) =119911119899+1

(2(119899 + 1)2

)

1119901

(119891 ∘ F) (119911)

=

119911119899+1119891 (119911

1 119911

119899)

(2(119899 + 1)2

)

1119901

(57)

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

For each 119904 gt minus1 we let V119904denote the measure on B

lowast

defined by

119889V119904(119911) = (1 minus 119873

2

lowast(119911))

119904

119889V (119911) (42)

where V denotes the normalized Lebesguemeasure onBlowast For

all 0 lt 119901 lt infin we define

A119901

119904(B

lowast) =H (B

lowast) cap 119871

119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) (43)

whereH(Blowast) is the space of all holomorphic functions onB

lowast

Naturally the spaces A119901

119904(B

lowast) are called weighted Bergman

spaces of Blowast

When 119901 = 2 there exists an orthogonal projection from1198712

(Blowast 119889V

119904) onto A2

119904(B

lowast) This will be called the weighted

Bergman projection and is denoted by P119904Blowast

It is well known that P

119904Blowast

is an integral operator on1198712

(Blowast 119889V

119904) namely

P119904Blowast

119891 (119911) = int

Blowast

K119904Blowast

(119911 119908) 119891 (119908) 119889V119904(119908) (44)

where

K119904Blowast

(119911 119908) =

A (119883 119884)(119899

2+ 119899 minus 119904) V

119904(B

lowast) (119883

2minus 119884)

119899+1+119904(45)

is the weighted Bergman kernel Here

119883 = 1 minus 119911 ∙ 119908 119884 = (119911 ∙ 119911)119908 ∙ 119908 (46)

and A(119883 119884) is the suminfin

sum

119895=0

119888119899119904119895119883119899+119904minus1minus2119895

119884119895

times [2 (119899 + 119904)119883 minus

(119899 minus 2119895 + 119904) (119899 + 1 + 2119904)

119899 + 119904 + 1

(1198832

minus 119884)]

(47)

As usual

119888119899119904119895= (

119899 + 119904 + 1

2119895 + 1)

=

(119899 + 119904 + 1) (119899 + 119904) sdot sdot sdot (119899 + 119904 minus 2119895 + 1)

(2119895 + 1)

(48)

is the binomial coefficient and V119904(B

lowast) is the weighted

Lebesgue volume of Blowast See [15 17] for these formulas and

more information about the weighted Bergman kernelsBecause of the infinite sum 119860(119883 119884) the formula for K

119904Blowast

is not really a closed form As such it is inconvenient for usto do estimates for P

119904Blowast

directlyThus we employ a techniquethat was used in [15] More specifically we relate the domainBlowastto the hypersurface M of the Euclidean unit ball in C119899+1

defined by

M = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 |119911| lt 1 (49)

If Pr C119899+1

rarr C119899 is defined by

Pr (1199111 119911

119899 119911

119899+1) = (119911

1 119911

119899) (50)

and F = Pr|M then F M rarr B

lowastminus 0 Let

H = 119911 isin C119899+1

0 119911 ∙ 119911 = 0 (51)

It was proved in [17] that there is an 119878119874(119899 + 1C)-invariantholomorphic form 120572 on H Moreover this form is uniqueup to a multiplicative constant In fact after appropriatenormalization the restriction to H cap (C 0)119899+1 of this formis given by

120572 (119911) =

119899+1

sum

119895=1

(minus1)119895minus1

119911119895

1198891199111and sdot sdot sdot and

119889119911

119895and sdot sdot sdot and 119889119911

119899+1 (52)

Our norm estimates for the weighted Bergman projectionP119904Blowast

on 119871119901 spaces will be based on the correspondingestimates onM Therefore we will consider the spaces 119871119901

119904(M)

consisting of measurable complex-valued functions 119891 on M

such that

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119904(M)= int

M

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119901

(1 minus |119911|2

)

119904120572 (119911) and 120572 (119911)

119862

lt infin (53)

where

119862 = (minus1)119899(119899+1)2

(2119894)119899

(54)

We use 119860119901119904(M) to denote the subspace of all holomorphic

functions in 119871119901119904(M) The weighted Bergman projection P

119904M

is then the orthogonal projection from 1198712119904(M) onto 1198602

119904(M)

Again it is well known that P119904M is an integral operator on

1198712

119904(M) given by the formula

P119904M119891 (119911) = int

M

K119904M (119911 119908) 119891 (119908) (1 minus |119908|

2

)

119904120572 (119911) and 120572 (119911)

119862

(55)

where K119904M is the corresponding Bergman kernel

The starting point for our analysis is the following closedform of K

119904M which was obtained as Theorem 32 in [15]

Lemma 11 (see [14]) The weighted Bergman kernel K119904M of

1198602

119904(M) is given by

K119904M (119911 119908) =

119862 (119899 minus 1 + (119899 + 1 + 2119904) 119911 ∙ 119908)

(1 minus 119911 ∙ 119908)119899+119904+1

(56)

where 119862 is a certain constant that depends on 119899 and 119904

Let 119891Blowastrarr C be a measurable function We define a

function T119891 onM by

(T119891) (119911) =119911119899+1

(2(119899 + 1)2

)

1119901

(119891 ∘ F) (119911)

=

119911119899+1119891 (119911

1 119911

119899)

(2(119899 + 1)2

)

1119901

(57)

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 7

The operator T will also play a key role in our analysis Inparticular we need the following result which was obtainedas Lemma 41 in [10]

Lemma 12 (see [14]) For each 119901 ⩾ 1 and 119904 gt minus1 thelinear operator T is an isometry from 119871119901(B

lowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

to 119871119901119904(M) Moreover we have P

119904MT = TP119904Blowast

on119871119901

(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904)

Theorem B of [15] states that the operator with positivekernel P

119904Blowast

maps 119871119901(Blowast |119911 ∙ 119911|

(119901minus2)2

119889V119904) boundedly onto

A119901

119904(B

lowast) for all 119901 gt 1 We need to derive an improved version

of the classical Forelli-Rudin integral estimates in the case ofthe minimal ball

Lemma 13 (see [14]) Suppose 119879 gt 0 119860 gt minus1 and 119889 isa nonnegative integer Then there exists a constant 119862 gt 0(depending on 119889 119879 and 119860 but not on 119905 and 119904) such that

int

M

|119911 ∙ 119908|2119889

|1 minus 119911 ∙ 119908|119899+119905+119904+1

(1 minus |119908|2

)

119904

120572 (119908) and 120572 (119908)

⩽

119862Γ (119904 + 1) Γ (119905)

(1 minus |119911|2

)

119905

(58)

for all minus1 lt 119904 lt 119860 0 lt 119905 lt 119879 and 119911 isin M

For any 119904 gt minus1 we consider the measure

119889120582119904(119911)

=

Γ (119899 + 119904)

2120596 (120597M) Γ (119899 minus 1) Γ (119904 + 1)(1 minus |119911|

2

)

119904

120572 (119911) and 120572 (119911)

(59)

onM where 119908 120596 is the (2119899 minus 1)-form on 120597M defined by

120596 (119911) (1198811 119881

2119899minus1) = 120572 (119911) and 120572 (119911) (119911 119881

1 119881

2119899minus1) (60)

Proposition 14 (see [14]) Suppose 1 le 119901 lt infin and minus1 lt 119904 ltinfin Then

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119901

le

1

(1 minus |119911|2

)

119899+1+119904int

M

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

119889120582119904(119908) (61)

for all 119892 isin A119901

119904(M) and 119911 isin M

This proposition leads directly to embeddings betweenAinfin

119904andA119901

1199041

for some 119904 and 1199041as we had above for unit disk

case (see [2]) and we showed that this repeating arguments ofunit disk case leads to new theorem we formulate below innext section for Bergman spaces in minimal ball

2 On Distances in 119870-Bergman Spaces LargeBergman-Orlicz Spaces in C119899 and BergmanSpaces in Minimal Ball

Based fully on preliminaries of previous section the plan ofthis section is the following first we formulate a result in the

unit disk and then repeat arguments we provided in proofof that theorem in various situations in higher dimensionsSince all proofs are short the repetitions of arguments whichare needed in higher dimensionwill be omitted only sketcheswill be given

It iswell known that119860119902minus119904119902minus1(U) sub 119860infin

minus119905(U) 119905 = 119904minus1119902 119905 lt

0 119904 lt 0 (see [18])It is easy to note based on previous section results that

the complete analogues of the embedding we just providedare valid also in Bergman spaces in minimal ball for 119899-harmonic function spaces in polydisk and in 119870-Bergmanspaces and this is based completely on preliminaries wehave in previous section We leave this easy task to readersThis allows us to pose a distance problem in each space weconsider in this paper

In the following theorem (see [1]) we calculate distancesfrom a weighted Bloch class to Bergman spaces for 119902 le 1 It iseasy to note that each argument below is also valid not only inunit disk but based on preliminaries in the previous sectionin Bergman spaces in minimal ball and polydisk and 119870-Bergman spaces and Bergman-Orlicz spaces So this theoremis very typical for us though it is known

Theorem 15 Let 0 lt 119902 le 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (1 minus119904119902)119902 minus 2 and 120573 gt minus1 minus 119905 Let 119891 isin 119860infin

minus119905 Then the following are

equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972= inf120576 gt 0 intU (intΩ

120576minus119905(119891)

((1 minus |119908|)120573+119905

|1minus

⟨119911 119908⟩|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

Proof First we show that 1198971le 119862119897

2 For 120573 gt minus1 minus 119905 we have

119891 (119911) = 119862 (120573)(int

UΩ120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908)

+int

Ω120576minus119905

119891 (119908) (1 minus |119908|)120573

(1 minus ⟨119908 119911⟩)120573+2

1198891198982(119908))

= 1198911(119911) + 119891

2(119911)

(62)

where119862(120573) is awell-knownBergman representation constant(see [5 18])

For 119905 lt 0

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862int

UΩ120576minus119905

1003816100381610038161003816119891 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576int

U

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908)

le 119862120576

1

(1 minus |119911|)minus119905

(63)

So sup119911isinU|1198911(119911)|(1 minus |119911|)

minus119905

lt 119862120576

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces

For 119904 lt 0 119905 lt 0 we have

int

U

10038161003816100381610038161198912(119911)1003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

Ω120576minus119905

(1 minus |119908|)120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911) le 119862

(64)

So we finally have

dist119860infin

minus119905

(119891 119860119902

minus119904119902minus1) le 119862

1003817100381710038171003817119891 minus 119891

2

1003817100381710038171003817119860infin

minus119905

= 11986210038171003817100381710038171198911

1003817100381710038171003817119860infin

minus119905

le 119862120576 (65)

It remains to prove that 1198972le 119897

1 Let us assume that

1198971lt 119897

2 Then we can find two numbers 120576 120576

1such that

120576 gt 1205761gt 0 and a function 119891

1205761

isin 119860119902

minus119904119902minus1 119891 minus 119891

1205761

119860infin

minus119905

le

1205761 and intU (intΩ

120576minus119905

((1 minus |119908|)120573+119905

|1 minus ⟨119911 119908⟩|120573+2

)1198891198982(119908))

119902

(1 minus

|119911|)minus119904119902minus1

1198891198982(119911) = infin Hence as above we easily get from

119891 minus 1198911205761

119860infin

minus119905

le 1205761that (120576 minus 120576

1)120594

Ω120576minus119905

(119891)(119911)(1 minus |119911|)

119905

le 119862|1198911205761

(119911)|and hence

119872 = int

U(int

U

120594Ω120576minus119905

(119891)(119911) (1 minus |119908|)

120573+119905

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

le 119862int

U(int

U

100381610038161003816100381610038161198911205761

(119908)

10038161003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119908 119911⟩|120573+2

1198891198982(119908))

119902

times (1 minus |119911|)minus119904119902minus1

1198891198982(119911)

(66)

Since for 119902 le 1 (see [5 18])

(int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816(1 minus |119911|)

120572

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905

)

119902

le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)120572119902+119902minus2

1198891198982(119911)

|1 minus ⟨119908 119911⟩|119905119902

(67)

where 120572 gt (1 minus 119902)119902 119905 gt 0 1198911205761

isin 119867(U) 119908 isin U and

int

U

(1 minus |119911|)minus119904119902minus1

|1 minus ⟨119908 119911⟩|119902(120573+2)

1198891198982(119911) le

119862

(1 minus |119908|)119902(120573+2)+119904119902minus1

(68)

where 119904 lt 0 120573 gt ((1 minus 119904119902)119902) minus 2 and 119908 isin U We get

119872 le 119862int

U

100381610038161003816100381610038161198911205761

(119911)

10038161003816100381610038161003816

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) (69)

So we arrive at a contradiction The theorem is proved

The following theorem is a version of Theorem 15 for thecase 119902 gt 1

Theorem 16 Let 119902 gt 1 119904 lt 0 119905 le 119904 minus 1119902 120573 gt (minus1 minus 119904119902)119902and 120573 gt minus1minus119905 Let119891 isin 119860infin

minus119905Then the following are equivalent

(a) 1198971= dist

119860infin

minus119905

(119891 119860119902

minus119904119902minus1)

(b) 1198972

= inf120576 gt 0 intU (intΩ120576minus119905

(119891)

((1 minus |119908|)120573+119905

|1 minus 119911119908|2+120573

)1198891198982(119908))

119902

(1 minus |119911|)minus119904119902minus1

1198891198982(119911) lt infin

The proof of Theorem 16 is the same actually as the proofof Theorem 15 The only difference is the boundedness ofBergman type projection operator but with positive Bergmankernel This fact will be used by us below heavily

Indeed the close inspection of the proof of Theorem 15shows that the proof of Theorem 16 is the same as the proofof Theorem 15 but here we will use (70) (see below) insteadof (67) For 120576 gt 0 119902 gt 1 120573 gt 0 120572 gt minus1119902 (see [18])

(int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

120572

|1 minus 119908119911|120573+2

1198891198982(119911))

119902

le 119862int

U

1003816100381610038161003816119891 (119911)

1003816100381610038161003816

119902

(1 minus |119911|)120572119902

|1 minus 119908119911|120573119902minus120576119902+2

1198891198982(119911) (1 minus |119908|)

minus120576119902

119908 isin U

(70)

which follows immediately fromHolderrsquos inequality and (68)(see [18])

We formulate one more theorem in this direction

Theorem 17 Let 120573119895gt 120573

0 119895 = 1 119899 for large enough 120573

0

119901 gt 1 and 120572119895gt 0 119895 = 1 119899 Then for 119891 isin ℎ(infininfin ) we

have

distℎ(infininfin)

(119891 ℎ (119901 119901 ))

≍ inf 120576 gt 0

int

U119899(int

Ω0

120576

119899

prod

119895=1

100381610038161003816100381610038161003816

119875120573119895

(119911119895 119908

119895)

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119908)

1003816100381610038161003816

times

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816)

120573119895+120572119895

1198891198982119899(119908))

119901

times

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)120572119896

1198891198982119899(119911) lt infin

(71)

where

Ω0

120576119905= 119911 isin U119899 100381610038161003816

1003816119891 (119911)

1003816100381610038161003816

119899

prod

119896=1

(1 minus1003816100381610038161003816119911119896

1003816100381610038161003816)

ge 120576

ge 0 120576 gt 0

(72)

Note also that now even a more general theorem forall mixed norm spaces is valid namely for ℎ(119901 119902 120572) spacesbased on facts we provided already above concerning projec-tion theorems We leave this to interested readers

The scheme of proofs of all high dimensional theorems ofthis section is the same The main tools are (see [1 2]) theBergman representation formula for larger space (for each119891 function from 119884 119891 isin 119884 119883 sub 119884) Then we use the factthat |119875|119891

119883le 119862119891

119883 119891 isin 119883 where |119875| is a Bergman

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 9

projectionwith positive Bergman kernelThenwe use Forelli-Rudin type estimate As example we provide a complete proofofTheorem 17 For a parallel proof of other assertionswe referthe reader to facts we provided above taken from mentionedpapers and leave this procedure to readers

Now we return to the proof of Theorem 17 We followarguments of unit disk case

Proof of Theorem 17 Let sup119903isin119868119899(prod

119899

119895=1(1 minus 119903

119895)120572119895

)119872infin(119891 119903) lt

infinThen intU119899 |119891(119911)|

119901

(prod119899

119895=1(1 minus |119911

119895|)120573119895

)1198891198982119899(119911) lt infin for all

120573119895gt 120573

0 119895 = 1 119899 where 120573

0is large enough This means

that based on lemmas on 119899-harmonic functions from theprevious section that for large enough 120573

119895120573119895gt1205730 119895 =

1 119899

119891 (119911) =

1

prod119899

119895=1Γ (120573119895)

times int

U119899(

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816)

120573119895minus1

)

times 119875 120573

(119911 120585) 119891 (120585) 1198891198982119899(120585) 119911 isin U119899

1003816100381610038161003816119875(119911 120585)

1003816100381610038161003816

le 119862 (120572 119899)

119899

prod

119895=1

1

100381610038161003816100381610038161 minus 120585

119895119911119895

10038161003816100381610038161003816

120572119895+1 119911 120585 isin U119899

120572119895ge 0

(73)

Then we have 120573119895gt1205730 119895 = 1 119899 119891 = 119891

1+ 119891

2as in unit

disk case

10038161003816100381610038161198911(119911)1003816100381610038161003816le 119862

1120576

1

prod119899

119895=1(1 minus

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816)

120572

10038161003816100381610038161198912(119911)1003816100381610038161003816le 119862

2

10038171003817100381710038171198911003817100381710038171003817ℎ(infininfin120572)

(74)

So finally we have distℎ(infininfin120572)

(1198912 ℎ(119901 120572)) le

119862119891 minus 1198912ℎ(infininfin120572)

= 1198621198911ℎ(infininfin120572)

To get the reverse we again repeat and follow arguments

of unit disk case and use the fact from the previous section

119878 1205730

119891 (119911)

= 119862 (1205730)int

U119899

119899

prod

119895=1

(1 minus

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

2

)

1205730minus1100381610038161003816100381610038161198751205730

(119911 120585)

10038161003816100381610038161003816119891 (120585) 119889119898

2119899(120585)

(75)

maps for large enough 1205730and for all 120572

119895gt 0 1 le 119901 119902 le

infin 119895 = 1 119899 119871(119901 119902 120572) into 119871(119901 119902 120572)

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 in 119870-Bergman spaces

Theorem 18 Let 120573 gt 1205730for large enough 120573

0 ≺ (lowast) 0 lt

119901 lt infin 119897120572= sum

119894gt0120572119894 Let 119892 isin 119867(B

119899) If |119875

120573|119892

119860119901

le

119862119892119860119901

hArr

int

B119899

(int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816(1 minus |119908|)

120573

|1 minus ⟨119911 119908⟩|119899+1+120573

119889] (119908))119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119911119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911)

le 119862int

B119899

1003816100381610038161003816119892 (119908)

1003816100381610038161003816

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119908)

(76)

then for 119891 isin 119860infin(2119899+119897120572)119901(B

119899) we have

dist119860infin

(2119899+119897120572)119901(B119899)(119891119867

119901

1205721120572119899

(B119899))

≍ inf

120576 gt 0 intB119899

(int

Ω1

120576119905(119891)

(1 minus |119908|)120573+119905

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)

119901

sdot

119899

prod

119894=1

(1 minus

119894

sum

119895=1

10038161003816100381610038161003816119908119895

10038161003816100381610038161003816

2

)

120572119894

119889] (119911) lt infin

(77)

where

Ω1

120576119905(119891) = 119911 isin B

119899 1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576

120576 gt 0 gt 0 =

2119899 + 119897120572

119901

(78)

Remark 19 Note that the condition in Theorem 18 is truewhen 120572

119895= 0 120572

119899= 0 or 120572

119895= 0 119895 = 1 119899 This is a classical

fact of theory of Bergman spaces in the unit ball (see [5])

The following theorem is based heavily on Remark 1which we provided above and Forelli-Rudin type estimatesfor unit ball (see [5] and Proposition 3) and some elementaryarguments we see in proof of unit disk case

We formulate complete analogue of Theorem 15 inBergman-Orlicz spaces

Theorem 20 Let 120572 gt minus1 0 lt 119901 lt infin Then for 119891 isin119860infin

(119899+1+120572)119901(B

119899) we have

dist119860infin

(119899+1+120572)119901(B119899)(119891 119860

Ψ119901

120572(B

119899))

≍ 120576 gt 0 int

B119899

Ψ119901(int

Ω2

120576(119899+1+120572)119901

(1 minus |119908|)120573minus(119899+1+120572)119901

119889] (119908)|1 minus ⟨119911 119908⟩|

119899+1+120573

)119889]120572

lt infin

(79)

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Function Spaces

where

Ω2

120576119905= 119911 isin B

1198991003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

gt 120576 120576 gt 0 gt 0

(80)

if |119875120573|119892

119871

Ψ119901

le 119862119892119871

Ψ119901

for 119892 isin 119860Ψ119901(B

119899)) gt minus1 120573 gt 120573

0

for large enough 1205730 where119875

120573is a weighted Bergman projection

in Bergman-Orlicz spaces

We formulate complete analogue of Theorem 15 in mini-mal ball

Theorem 21 Let 120573 gt 1205730for large enough 120573

0 1 lt 119901 lt

infin minus1 lt 119904 lt infin Then for any 119891 isin 119860infin(119899+1+119904)119901

(119872) for whichBergman representation (55) is valid

dist119860infin

(119899+1+119904)119901(119872)(119891 119860

119901

119904(119872))

≍ inf 120576 gt 0 int119872

(int

Ω3

120576(119899+1+119904)119901

10038161003816100381610038161003816119870120573119872(119911 119908)

10038161003816100381610038161003816119891 (119908)

times (1 minus |119908|)120573+119905

119889120582 (119908))

119901

119889120582119904(119908) lt infin

(81)

where 119905 = minus(119899 + 1 + 119904)119901 and

Ω3

120576119905= 119911 isin 119872

1003816100381610038161003816119891 (119911)

1003816100381610038161003816(1 minus |119911|)

ge 120576 gt 0 120576 gt 0 (82)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by MNTR Serbia Project 174017

References

[1] R Shamoyan and O R Mihic ldquoOn new estimates for distancesin analytic function spaces in the unit disk the polydisk and theunit ballrdquo Boletın de la Asociacion Matematica Venezolana vol17 no 2 pp 89ndash103 2010

[2] R F Shamoyan and O R Mihic ldquoOn new estimates fordistances in analytic function spaces in higher dimensionrdquoSiberian Electronic Mathematical Reports vol 6 pp 514ndash5172009

[3] R F Shamoyan ldquoOn an extremal problem in analytic functionspaces in tube domains over symmetric conesrdquo to appear inIssues of Analysis

[4] R F Shamoyan ldquoOn an extremal problem in analytic spaces intwo Siegel domains inC119899rdquoROMAI Journal vol 8 no 2 pp 167ndash180 2012

[5] F A Shamoian and M Djrbashian Topics in the Theory ofBergman A119901

120572Spaces Teubner-Texte zur Mathematik Teubner

Leipzig Germany 1988

[6] A Karapetyan ldquoWeighted anisotropic integral representationsof holomorphic functions in the unit ball of C119899rdquo Abstract andApplied Analysis vol 2010 Article ID 354961 23 pages 2010

[7] F Forelli and W Rudin ldquoProjections on spaces of holomorphicfunctions in ballsrdquo Indiana University Mathematics Journal vol24 pp 593ndash602 1975

[8] M Djrbashian ldquoSurvey of some achievements of Armenianmathematicians in the theory of integral representations andfactorization of analytic functionsrdquoMatematicki Vesnik vol 39no 3 pp 263ndash282 1987

[9] M Djrbashian ldquoA brief survey of the results of researchby Armenian mathematicians in the theory of factorizationof meromorphic functions and its applicationsrdquo Journal ofContemporary Mathematical Analysis vol 23 no 6 pp 8ndash341988

[10] S Charpentier ldquoComposition operators on weighted Bergman-Orlicz spaces on the ballrdquo Complex Analysis and OperatorTheory vol 7 no 1 pp 43ndash68 2011

[11] S Charpentier and B Sehba ldquoCarleson measure theoremsfor large Hardy-Orlicz and Bergman-Orlicz spacesrdquo Journal ofFunction Spaces and Applications vol 2012 Article ID 79276321 pages 2012

[12] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[13] K L Avetisyan ldquoWeighted spaces of harmonic and holomor-phic functionsrdquo Armenian Journal of Mathematics vol 2 no 42009

[14] J Gonessa and K Zhu ldquoA sharp norm estimate for weightedBergman projections on the minimal ballrdquo New York Journal ofMathematics vol 17a pp 101ndash112 2011

[15] G Mengotti and E H Youssfi ldquoThe weighted Bergman pro-jection and related theory on the minimal ballrdquo Bulletin desSciences Mathematiques vol 123 no 7 pp 501ndash525 1999

[16] K T Hahn and P Pflug ldquoOn a minimal complex norm thatextends the real Euclidean normrdquoMonatshefte fur Mathematikvol 105 no 2 pp 107ndash112 1988

[17] K Oeljeklaus and EH Youssfi ldquoProper holomorphicmappingsand related automorphism groupsrdquo The Journal of GeometricAnalysis vol 7 no 4 pp 623ndash636 1997

[18] JMOrtega and J Fabrega ldquoHardyrsquos inequality and embeddingsin holomorphic Triebel-Lizorkin spacesrdquo Illinois Journal ofMathematics vol 43 no 4 pp 733ndash751 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of