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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction and Main Definitions

Let Open image in new window and Open image in new window be the Open image in new window -dimensional space of complex coordinates. We denote the unit polydisk by
and the distinguished boundary of Open image in new window by
We use Open image in new window to denote the volume measure on Open image in new window and Open image in new window to denote the normalized Lebesgue measure on Open image in new window Let Open image in new window be the space of all holomorphic functions on Open image in new window When Open image in new window we simply denote Open image in new window by Open image in new window Open image in new window by Open image in new window Open image in new window by Open image in new window Open image in new window by Open image in new window We refer to [1, 2] for further details. We denote the expanded disk by
and the subframe by
The Hardy spaces, denoted by Open image in new window are defined by

where Open image in new window Open image in new window

For Open image in new window recall that the weighted Bergman space Open image in new window consists of all holomorphic functions on the polydisk satisfying the condition
For Open image in new window the Bergman class on expanded disk is defined by
and similarly the Bergman class on subframe denoted by Open image in new window is defined by

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

We use standard covering Lemma A to construct a set Open image in new window such that Open image in new window - boxes Open image in new window pairwise disjoint, we have
So the lemma will be proved if Open image in new window Let

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

The last estimate follows from inclusion Open image in new window
It remains to show the inclusion Open image in new window To show this inclusion we note if Open image in new window then we have Open image in new window Using covering Lemma A we have Open image in new window Hence
It remains to note that we put above Open image in new window
  1. (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.

Note that one part of the theorem was proved in [3] and follows from Theorem A. Let us show the reverse. Let first Open image in new window Consider the following Open image in new window function:
where Open image in new window can be large enough, Open image in new window is a constant of Bergman from representation formula (see [1]), obviously Open image in new window by Bergman representation formula in the unit disk. It remains to note that the following estimate hold by Lemma 2.3:

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

by Bergman representation formula. Obviously Open image in new window , as
where Open image in new window is a map from subframe Open image in new window to diagonal. Using duality arguments and Fubini's theorem we have

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

Indeed using Hölder's inequality we get
where Open image in new window We used above the estimate
Returning to estimate for Open image in new window we have by Hölder's inequality

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.

For Open image in new window we have to use again Lemma 2.3 and Fubini's theorem. For Open image in new window we first prove Open image in new window if Open image in new window and if Open image in new window
Indeed by Hölder's inequality we have

Hence calculating integrals we finally have Open image in new window

We used the estimate

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.

Note that estimates between expanded disk, unit disk, and polydisk can be also obtained directly from Liuville's formula

Remark 3.6.

The complete description of traces of classes with Open image in new window and Open image in new window quasinorms on the unit disk can be obtained similarly by small modification of the proof of Theorem 3.2,

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

Note that one part of statement is obvious. If, for example, Open image in new window then Open image in new window On the other side, let Open image in new window Then define as above that
is big enough, Open image in new window is a Bergman constant of Bergman representation formula.

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.

We have for such a function

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

In the case of the unit ball an analogue of Open image in new window operator is a well-known radial derivative which is well studied. We note that in polydisk the following fractional derivative is well studied (see [1]):
where Open image in new window Open image in new window , and Open image in new window Apparently the Open image in new window operator was studied in [12] for the first time. Then in [13], the second author studied some properties of this operator. In this section we also continue to study the Open image in new window operator. We need the following simple but vital formula which can be checked by easy calculation:

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

The following dyadic decomposition of subframe and polydisk was introduced in [1] and will be also used by us:
averages in analytic spaces in polydisk can obviously have a mixed form, for example, Open image in new window In [8] Ren and Shi described the diagonal of mixed norm spaces, but the above mentioned mixed case was omitted there. Our approach is also different. It is based on dyadic decomposition we introduced previously.

Theorem 4.1.

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

Proof.

Using diadic decomposition of polydisk we have
We used above the following estimate which can be found, for example, in [1]
where Open image in new window are enlarged dyadic cubes (see [1]) and Open image in new window , and

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.

To get the reverse statement we use the estimate from Lemma 2.3. Then we have
Let Open image in new window Then we may assume that again Open image in new window is large enough. Let Open image in new window Then
We used estimate

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]).

In [14] Carleson as Rudin and Clark showed that in the case of the polydisk one cannot expect so simple description of Carleson measures as one has for measures defined in the disk. We would like to study embeddings of the type

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.

and conversely if

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.

It can be checked by direct calculations based on formula (4.2) that the following integral representation holds:
Using the fact that Open image in new window in increasing by Open image in new window and Open image in new window (see [1]) we get from above by using the estimate
To obtain the reverse implication we use standard test function Open image in new window and Lemma 2.5

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 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 there exists "an extension" Open image in new window such that
The following Theorem 4.6 gives an answer for the same map from polydisk to expanded disk

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.

Let Open image in new window be a positive Borel measure on Open image in new window Then

if and only if Open image in new window

Proof.

Obviously if Open image in new window then by putting

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.

Let Open image in new window

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]).

Let Open image in new window We will show now as in [4] that
Indeed, this will be enough, since the operator Open image in new window is Open image in new window operator we can apply the Marcinkiewicz interpolation theorem (see [16, Chapter 1]) to assert that

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

Indeed by Proposition 2.1 we have

Theorem 4.6 is proved.

Theorem 4.7.

where Open image in new window

Proof.

We use systematically the integral representation (4.18), Lemma 2.5, and it is corollary. The proof of the estimate (4.32) follows from equality
obtained during the proof of Theorem 4.3.

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

The proof of the estimate (4.34) follows from equality Open image in new window

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]).

We present below a complete analogue of Theorems 3.2 and 3.3 for a map from polydisk to expanded disk. Note the continuation of Open image in new window function is done again from diagonal Open image in new window Let Open image in new window and

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:

For any function Open image in new window with a finite quasinorm Open image in new window such that condition (4.38) holds for Open image in new window one has Open image in new window

Proof.

Proof of estimate (4.39) follows directly from Theorem B.
Indeed from (4.38) and results of [1] on diagonal map in Bergman classes we have Open image in new window It remains to apply Theorem A.
For the proof of Open image in new window we use Theorem 4.6 and get the result we need.

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

The proof of first part of Proposition 4.11 follows from (4.35), (4.36) and Lemma 2.5 directly. The reverse assertion follows from Theorem B Open image in new window and estimate

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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Copyright information

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

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsBryansk UniversityBryanskRussia
  2. 2.Faculty of Organizational SciencesUniversity of BelgradeBelgradeSerbia

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