- Romi F. Shamoyan
^{1}and - Olivera R. Mihić
^{2}Email author

**2009**:353801

**DOI: **10.1155/2009/353801

© R. F. Shamoyan and O. R. Mihić. 2009

**Received: **25 November 2008

**Accepted: **28 September 2009

**Published: **11 October 2009

## Abstract

We introduce and study new analytic classes on subframe and expanded disk and give complete description of their traces on the unit disk. Sharp embedding theorems and various new estimates concerning differential operator in polydisk also will be presented. Practically all our results were known or obvious in the unit disk.

## 1. Introduction and Main Definitions

Throughout the paper, we write (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables being discussed.

The notation means that there is a positive constant such that We will write for two expressions if there is a positive constant such that

This paper is organized as follows. In first section we collect preliminary assertions. In the second section we present several new results connected with so-called operator of diagonal map in polydisk. Namely, we define two new maps and from subframe and expanded disk to unit disk and, in particular, completely describe traces of Bergman classes and defined on subframe and expanded disk on usual unit disk on the complex plane. Proofs are based among other things on new projection theorems for these classes.

A separate section will be devoted to the study of differential operator in polydisk. It is based in particular on results from the recent paper [3]. We will use the dyadic decomposition technique to explore connections between analytic classes on subframe, polydisk, and expanded disk. We also prove new sharp embedding theorems for classes on subframe and expanded disk. Last assertions of the final section generalize some one-dimensional known results to polydisk and to the case of operators simultaneously.

## 2. Preliminaries

We need the following assertions.

Lemma 2 A (see [4]).

Let be a fixed -tuple of nonnegative numbers and let be an arbitrary family of -boxes in lying in the cube There exists a set such that and for all there exists such that

The following proposition is heavily based on ideas from [4].

Proposition 2.1.

Proof.

Proofs of all parts are similar. The first part of the lemma connected with measure and unit disk can be found in [4]. We will give the complete proof of the second part. To prove the second part let

Let then we will show So where , and constant will be specified later. Hence we will have

Now part (b) can be obtained by direct calculation.

Remark 2.2.

In Proposition 2.1(b) can be replaced by

Lemma 2.3.

Estimate (2.11) for can be found in [5] and in [1] for general case. The following lemma is well known.

Lemma 2.4.

Let Then for one has (a) ; (b) ; (c)

Lemma 2.5 (see [3]).

Corollary 2.6.

We will need the following Theorems http://A and http://B.

Theorem 2 A (see [3]).

Theorem 2 B (see [3]).

## 3. Analytic Classes on Subframe and Expanded Disk

Let us remind the main definition.

Definition 3.1.

Let be subspaces of and We say that the diagonal of coincides with if for any function , and the reverse is also true for every function from there exists an expansion such that Then we write

where is an arbitrary analytic expansion of from diagonal of polydisk to polydisk.

The problem of study of diagonal map and its applications for the first time was also suggested by Rudin in [2]. Later several papers appeared where complete solutions were given for classical holomorphic spaces such as Hardy, Bergman classes; see [1, 4, 6, 7] and references there. Recently the complete answer was given for so-called mixed norm spaces in [8]. Partially the goal of this paper is to add some new results in this direction. Theorems on diagonal map have numerous applications in the theory of holomorphic functions (see, e.g., [9, 10]).

In this section we concentrate on the study of two maps closely connected with diagonal mapping from subframe into disk where all and another map from expanded disk into disk where and function is from a functional class on subframe or expanded disk

Note that the study of maps which are close to diagonal mapping was suggested by Rudin in [2] and previously in [11] Clark studied such a map.

Theorem 3.2.

(b) then for every function and for every function there exist such that

Proof.

where can be large enough. Using Fubini's theorem and calculating the inner integral we get what we need.

We will need the following assertion. Let

where We assert that belongs to

We used the fact that for all proved before.

The complete analogue of Theorem 3.2 is true for Bergman classes on expanded disk we defined previously.

Theorem 3.3.

(b) then the following assertion holds. For every function belongs to and the reverse is also true, for any function from there exists a function such that for all

Proof.

We give a short sketch of proof of Theorem 3.3 and omit details. Note that the half of the theorem the inclusion was proved in [3] and follows directly from Theorem A.

Hence calculating integrals we finally have

which is true under the conditions on indexes we have in formulation of theorem and can be obtained by using Hölder's inequality for functions. Using this projection theorem and repeating arguments of proof of the previous theorem we will complete the proof of Theorem 3.3.

Remark 3.4.

Note that Theorems 3.2 and 3.3 are obvious for

Remark 3.5.

Remark 3.6.

We formulate complete analogues of Theorems 3.2 and 3.3 for classes

Theorem 3.7.

Let and Then is in and any can be expanded to such that The same statement is true for pairs

Obviously Using Hölder's inequality for functions we get Similarly

It is natural to question about discrete analogues of operators we considered previously.

As a consequence of these arguments and using Lemma 2.4 we have the following proposition, a discrete copy of assertions we proved above.

Proposition 3.8.

## 4. Sharp Embeddings for Analytic Spaces in Polydisk with Operators and Inequalities Connecting Classes on Polydisk, Subframe, and Expanded Disk

The goal of this section is to present various generalizations of well-known one-dimensional results providing at the same time new connections between standard classes of analytic functions with quazinorms on polydisk and differential operator with corresponding classes on subframe and expanded disk.

In this section we also study another two maps connected with the diagonal mapping from polydisk to subframe and expanded disk using, in particular, estimates for maximal functions from Lemma 2.3 which are of independent interest. Note that for the first time the study of such mappings which are close to diagonal mapping was suggested by Rudin in [2]. Later Clark studied such a map in [11].

In this section we also introduce the differential operator as follows (see [3, 12, 13]). where

Note it is easy to check that acts from into

where This simple integral representation of holomorphic functions in polydisk will allow us to consider them in close connection with functional spaces on subframe

Theorem 4.1.

Proof.

Hence using the fact that is a finite covering of finally we have for all One part of theorem is proved.

Choosing appropriate we repeat now arguments that we presented for above to get what we need. The proof is complete.

Remark 4.2.

The case of averages can be considered similarly. Note in Theorem 4.1 Thus our Theorem 4.1 extends known description of diagonal of classical Bergman classes (see [1, 7]).

where is a positive Borel measure on and is a Bergman class on polydisk or subframe.

Theorem 4.3.

Remark 4.4.

With another -sharp" embedding theorem, the complete analogue of Theorem 4.3 is true also when we replace the left side by The proof needs small modification of arguments we present in the proof of Theorem 4.3.

Remark 4.5.

For and Theorem 4.3 is known (see [15]).

Proof of Theorem 4.3.

The rest is clear. The proof is complete.

Below we continue to study connections between standard classes in polydisk and corresponding spaces on subframe and expanded disk.

Let us note that Theorems 3.2 and 3.7 of [3] show that under some restrictions on the following assertion is true.

We will develop ideas from [4] to get the following sharp embedding theorem for classes on expanded disk.

Theorem 4.6.

Proof.

we have and Hence we get what we need. Now we will show the sufficiency of the condition.

In what follows we will use notations of Proposition 2.1. Consider the Poisson integral of a function (see [2]).

where we used the standard partition of Poisson integral. Hence and so using Proposition 2.1 we finally get

Theorem 4.6 is proved.

Theorem 4.7.

Proof.

The proof of the estimate (4.33) follows from Lemma 2.5 and its corollary and integral representation (4.35).

Indeed using (4.36) integrating both sides of (4.37) by and using Lemma 2.5 we arrive at (4.34).

Remark 4.8.

All estimates in Theorem 4.7 for are well known (see [1, Chapter1]).

where is a constant of Bergman representation formula (see [1]).

Proposition 4.9.

And reverse is also true:

then for all functions such that condition (4.38) holds for we have

and the reverse is also true:

Proof.

For the proof of we use the same argument as in the proof of part (2.11). Namely first from (4.38) and from a Diagonal map theorem on classes from [1] we get It remains to apply Theorem A.

Remark 4.10.

Note Proposition 4.9 is obvious for

We give only a sketch of the proof of the following result. It is based completely on a technique we developed above.

Proposition 4.11.

which can obtained from one dimensional result by induction.

The proof of second part of Proposition 4.11 can be obtained from Theorem A and results on diagonal map on Hardy classes from [1] similarly as the proof of Proposition 4.9. For part is well known (e.g., see [1]) and follows from since for condition (4.38) vanishes by Bergman representation formula.

Remark 4.12.

Theorem 4.6, Propositions 4.9 and 4.11 give an answer to a problem of Rudin (see [2]) to find traces of Hardy classes on subvarieties other than diagonal Note that in [11] Clark solved this problem for subvarietes of based on finite Blaschke products.

## Declarations

### Acknowledgment

The authors sincerely thank Trieu Le for valuable discussions.

## Authors’ Affiliations

## References

- Djrbashian AE, Shamoian FA:
*Topics in the Theory of Apα Spaces, Teubner Texts in Mathematics*.*Volume 105*. Teubner, Leipzig, Germany; 1988.Google Scholar - Rudin W:
*Function Theory in Polydiscs*. W. A. Benjamin, New York, NY, USA; 1969:vii+188.MATHGoogle Scholar - Shamoyan R, Li S:
**On some properties of a differential operator on the polydisk.***Banach Journal of Mathematical Analysis*2009,**3**(1):68–84.MathSciNetView ArticleMATHGoogle Scholar - Shamoian FA:
**Embedding theorems and characterization of traces in the spaces .***Matematicheskii Sbornik*1978,**107**(3):709–725.Google Scholar - Ortega JM, Fàbrega J:
**Hardy's inequality and embeddings in holomorphic Triebel-Lizorkin spaces.***Illinois Journal of Mathematics*1999,**43**(4):733–751.MathSciNetMATHGoogle Scholar - Jevtić M, Pavlović M, Shamoyan RF:
**A note on the diagonal mapping in spaces of analytic functions in the unit polydisc.***Publicationes Mathematicae Debrecen*2009,**74**(1–2):45–58.MathSciNetMATHGoogle Scholar - Shapiro JH:
**Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces.***Duke Mathematical Journal*1976,**43**(1):187–202. 10.1215/S0012-7094-76-04316-7MathSciNetView ArticleMATHGoogle Scholar - Ren G, Shi J:
**The diagonal mapping in mixed norm spaces.***Studia Mathematica*2004,**163**(2):103–117. 10.4064/sm163-2-1MathSciNetView ArticleMATHGoogle Scholar - Amar E, Menini C:
**A counterexample to the corona theorem for operators on .***Pacific Journal of Mathematics*2002,**206**(2):257–268. 10.2140/pjm.2002.206.257MathSciNetView ArticleMATHGoogle Scholar - Shamoyan R:
**On the action of Hankel operators in bidisk and subspaces in connected with inner functions in the unit disk.***Comptes Rendus de l'Académie Bulgare des Sciences*2007,**60**(9):929–934.MathSciNetMATHGoogle Scholar - Clark DN:
**Restrictions of functions in the polydisk.***American Journal of Mathematics*1988,**110**(6):1119–1152. 10.2307/2374688MathSciNetView ArticleMATHGoogle Scholar - Guliev VS, Lizorkin PI:
**Classes of holomorphic and harmonic functions in a polydisk in connection with their boundary values.***Trudy Matematicheskogo Instituta imeni V. A. Steklova*1993,**204:**137–159.MathSciNetMATHGoogle Scholar - Shamoyan RF: On quasinorms of functions from holomorphic Lizorkin Triebel type spaces on subframe and polydisk. In Sympozium Fourier Series and Application, Conference Materials, 2002. Rostov Na Donu; 54–55.Google Scholar
- Carleson L:
*A counter example for measures bounded in for the bidisc.*Institut Mittag-Leffler; 1974.Google Scholar - Wu Z:
**Carleson measures and multipliers for Dirichlet spaces.***Journal of Functional Analysis*1999,**169**(1):148–163. 10.1006/jfan.1999.3490MathSciNetView ArticleMATHGoogle Scholar - Mazya VG:
*Sobolev Spaces, Springer Series in Soviet Mathematics*. Springer, Berlin, Germany; 1985:xix+486.Google Scholar

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