© R. F. Shamoyan and O. R. Mihić. 2009
Received: 25 November 2008
Accepted: 28 September 2009
Published: 11 October 2009
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.
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 . 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.
We need the following assertions.
Lemma 2 A (see ).
The following proposition is heavily based on ideas from .
Proofs of all parts are similar. The first part of the lemma connected with measure and unit disk can be found in . We will give the complete proof of the second part. To prove the second part let
Now part (b) can be obtained by direct calculation.
Lemma 2.5 (see ).
Theorem 2 A (see ).
Theorem 2 B (see ).
3. Analytic Classes on Subframe and Expanded Disk
Let us remind the main definition.
The problem of study of diagonal map and its applications for the first time was also suggested by Rudin in . 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 . 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
The complete analogue of Theorem 3.2 is true for Bergman classes on expanded disk we defined previously.
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  and follows directly from Theorem A.
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.
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.
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 . Later Clark studied such a map in .
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.
For and Theorem 4.3 is known (see ).
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  show that under some restrictions on the following assertion is true.
We will develop ideas from  to get the following sharp embedding theorem for classes on expanded disk.
In what follows we will use notations of Proposition 2.1. Consider the Poisson integral of a function (see ).
Theorem 4.6 is proved.
The proof of the estimate (4.33) follows from Lemma 2.5 and its corollary and integral representation (4.35).
All estimates in Theorem 4.7 for are well known (see [1, Chapter1]).
where is a constant of Bergman representation formula (see ).
And reverse is also true:
and the reverse is also true:
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  we get It remains to apply Theorem A.
We give only a sketch of the proof of the following result. It is based completely on a technique we developed above.
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  similarly as the proof of Proposition 4.9. For part is well known (e.g., see ) and follows from since for condition (4.38) vanishes by Bergman representation formula.
Theorem 4.6, Propositions 4.9 and 4.11 give an answer to a problem of Rudin (see ) to find traces of Hardy classes on subvarieties other than diagonal Note that in  Clark solved this problem for subvarietes of based on finite Blaschke products.
The authors sincerely thank Trieu Le for valuable discussions.
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