# Analytic Classes on Subframe and Expanded Disk and the Open image in new window Differential Operator in Polydisk

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## 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 Open image in new window operator in polydisk also will be presented. Practically all our results were known or obvious in the unit disk.

### Keywords

Holomorphic Function Unit Disk Bergman Space Carleson Measure Dyadic Cube## 1. Introduction and Main Definitions

where Open image in new window Open image in new window

where Open image in new window

Throughout the paper, we write Open image in new window (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 Open image in new window means that there is a positive constant Open image in new window such that Open image in new window We will write for two expressions Open image in new window if there is a positive constant Open image in new window such that Open image in new window

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 Open image in new window and Open image in new window from subframe Open image in new window and expanded disk Open image in new window to unit disk Open image in new window and, in particular, completely describe traces of Bergman classes Open image in new window and Open image in new window defined on subframe and expanded disk on usual unit disk Open image in new window 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 Open image in new window 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 Open image in new window operators simultaneously.

## 2. Preliminaries

We need the following assertions.

Lemma 2 A (see [4]).

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

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 Open image in new window 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 Open image in new window

Let Open image in new window then we will show Open image in new window Open image in new window So Open image in new window where Open image in new window , and constant Open image in new window will be specified later. Hence we will have Open image in new window

- (b)Note that for the case of Open image in new window we can step by step repeat the same procedure with Open image in new window instead of Open image in new window and the condition on Open image in new window will be replaced by weaker condition

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

Remark 2.2.

In Proposition 2.1(b) Open image in new window can be replaced by Open image in new window

Lemma 2.3.

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

Lemma 2.4.

Let Open image in new window Then for Open image in new window one has (a) Open image in new window ; (b) Open image in new window ; (c) Open image in new window

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

where Open image in new window is an arbitrary analytic expansion of Open image in new window 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 Open image in new window from subframe into disk Open image in new window where all Open image in new window and another map Open image in new window from expanded disk into disk where Open image in new window and Open image in new window function is from a functional class on subframe Open image in new window or expanded disk Open image in new window

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.

Let Open image in new window If

(a) Open image in new window or

(b) Open image in new window then for every function Open image in new window and for every function Open image in new window there exist Open image in new window such that Open image in new window

Proof.

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

where Open image in new window

We will need the following assertion. Let Open image in new window

where Open image in new window We assert that Open image in new window belongs to Open image in new window

We used the fact that for all Open image in new window proved before.

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

Theorem 3.3.

Let Open image in new window If

(a) Open image in new window or

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

Proof.

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

Hence calculating integrals we finally have Open image in new window

which is true under the conditions on indexes we have in formulation of theorem and can be obtained by using Hölder's inequality for Open image in new window 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 Open image in new window

Remark 3.5.

Remark 3.6.

We formulate complete analogues of Theorems 3.2 and 3.3 for classes Open image in new window

Theorem 3.7.

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

Obviously Open image in new window Using Hölder's inequality for Open image in new window functions we get Open image in new window Similarly Open image in new window

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.

Let Open image in new window and Open image in new window Then Open image in new window if and only if Open image in new window Open image in new window Open image in new window

## 4. Sharp Embeddings for Analytic Spaces in Polydisk with Open image in new window 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 Open image in new window 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 Open image in new window differential operator as follows (see [3, 12, 13]). Open image in new window where Open image in new window

Note it is easy to check that Open image in new window acts from Open image in new window into Open image in new window

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

Theorem 4.1.

Let Open image in new window and Open image in new window Then Open image in new window

Proof.

We used the fact that Open image in new window

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

Choosing appropriate Open image in new window we repeat now arguments that we presented for Open image in new window above to get what we need. The proof is complete.

Remark 4.2.

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

where Open image in new window is a positive Borel measure on Open image in new window and Open image in new window is a Bergman class on polydisk or subframe.

Theorem 4.3.

Remark 4.4.

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

Remark 4.5.

For Open image in new window and Open image in new window Theorem 4.3 is known (see [15]).

Proof of Theorem 4.3.

where Open image in new window

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 Open image in new window the following assertion is true.

for Open image in new window functions from Open image in new window

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

Theorem 4.6.

if and only if Open image in new window

Proof.

we have Open image in new window and Open image in new window Open image in new window 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 Open image in new window (see [2]).

where we used the standard partition of Poisson integral. Hence Open image in new window and so using Proposition 2.1 we finally get Open image in new window

Theorem 4.6 is proved.

Theorem 4.7.

where Open image in new window

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 Open image in new window and using Lemma 2.5 we arrive at (4.34).

Remark 4.8.

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

where Open image in new window is a constant of Bergman representation formula (see [1]).

Proposition 4.9.

And reverse is also true:

then for all functions Open image in new window such that condition (4.38) holds for Open image in new window we have Open image in new window

and the reverse is also true:

Proof.

For the proof of Open image in new window 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 Open image in new window classes from [1] we get Open image in new window It remains to apply Theorem A.

Remark 4.10.

Note Proposition 4.9 is obvious for Open image in new window

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.

If Open image in new window then Open image in new window

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 Open image in new window from [1] similarly as the proof of Proposition 4.9. For Open image in new window part Open image in new window is well known (e.g., see [1]) and Open image in new window follows from Open image in new window since for Open image in new window 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 Open image in new window Hardy classes on subvarieties other than diagonal Open image in new window Note that in [11] Clark solved this problem for subvarietes of Open image in new window based on finite Blaschke products.

## Notes

### Acknowledgment

The authors sincerely thank Trieu Le for valuable discussions.

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