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Analytic Classes on Subframe and Expanded Disk and the
Differential Operator in Polydisk
Journal of Inequalities and Applications volume 2009, Article number: 353801 (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
Let and
be the
-dimensional space of complex coordinates. We denote the unit polydisk by

and the distinguished boundary of by

We use to denote the volume measure on
and
to denote the normalized Lebesgue measure on
Let
be the space of all holomorphic functions on
When
we simply denote
by
by
by
by
We refer to [1, 2] for further details. We denote the expanded disk by

and the subframe by

The Hardy spaces, denoted by are defined by

where
For recall that the weighted Bergman space
consists of all holomorphic functions on the polydisk satisfying the condition

For the Bergman class on expanded disk is defined by

and similarly the Bergman class on subframe denoted by is defined by

where
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.

Let be a nonnegative summable function on
Let
Let
where
and where
is a family of all
boxes such that
is proportional to
for some fixed
Then the following statements hold:

where and
are any positive Borel measures on
and
such that

where (b) Let
Then

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
Fix a point in the
and associate an
- box
such that

We use standard covering Lemma A to construct a set such that
- boxes
pairwise disjoint, we have

So the lemma will be proved if Let

Let then we will show
So
where
, and constant
will be specified later. Hence we will have
The last estimate follows from inclusion

It remains to show the inclusion To show this inclusion we note if
then we have
Using covering Lemma A we have
Hence

It remains to note that we put above
-
(b)
Note that for the case of
we can step by step repeat the same procedure with
instead of
and the condition on
will be replaced by weaker condition
(29)
Note for

Now part (b) can be obtained by direct calculation.
Remark 2.2.
In Proposition 2.1(b) can be replaced by
Lemma 2.3.
Let Then

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]).
Let Then one has

Corollary 2.6.
Let Then

We will need the following Theorems http://A and http://B.
Theorem 2 A (see [3]).

Let Then


Let Then

Theorem 2 B (see [3]).

Let Then


Let Then

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
Note when then

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.
Let If
(a) or
(b) then for every function
and for every function
there exist
such that
Proof.
Note that one part of the theorem was proved in [3] and follows from Theorem A. Let us show the reverse. Let first Consider the following
function:

where can be large enough,
is a constant of Bergman from representation formula (see [1]), obviously
by Bergman representation formula in the unit disk. It remains to note that the following estimate hold by Lemma 2.3:

where can be large enough. Using Fubini's theorem and calculating the inner integral we get what we need.
We consider now case. Let
Then

by Bergman representation formula. Obviously , as

where is a map from subframe
to diagonal. Using duality arguments and Fubini's theorem we have

where
We will need the following assertion. Let
Then let

where We assert that
belongs to
Indeed using Hölder's inequality we get

where We used above the estimate


Returning to estimate for we have by Hölder's inequality

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.
Let If
(a) or
(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.
For we have to use again Lemma 2.3 and Fubini's theorem. For
we first prove
if
and if

Indeed by Hölder's inequality we have

Hence calculating integrals we finally have
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 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.
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 and
quasinorms on the unit disk can be obtained similarly by small modification of the proof of Theorem 3.2,

Let

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
Note that one part of statement is obvious. If, for example, then
On the other side, let
Then define as above that


is big enough, is a Bergman constant of Bergman representation formula.
Obviously Using Hölder's inequality for
functions we get
Similarly
It is natural to question about discrete analogues of operators we considered previously.
Let Let

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 and
Then
if and only if
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
In the case of the unit ball an analogue of 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 , and
Apparently the
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
operator. We need the following simple but vital formula which can be checked by easy calculation:

where This simple integral representation of holomorphic
functions in polydisk will allow us to consider them in close connection with functional spaces on subframe
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, 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 and
Then
Proof.
Using diadic decomposition of polydisk we have

We used above the following estimate which can be found, for example, in [1]

where are enlarged dyadic cubes (see [1]) and
, and

Note further since

We have

We used the fact that
Hence using the fact that is a finite covering of
finally we have for all
One part of theorem is proved.
To get the reverse statement we use the estimate from Lemma 2.3. Then we have

for any Hence
and for
by Lemma 2.3

Let Then we may assume that again
is large enough. Let
Then

We used estimate

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]).
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 is a positive Borel measure on
and
is a Bergman class on polydisk or subframe.
Theorem 4.3.
Let , and
is a positive Borel measure on
If

then

and conversely if

holds for some where
then

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.
It can be checked by direct calculations based on formula (4.2) that the following integral representation holds:


Using the fact that in increasing by
and
(see [1]) we get from above by using the estimate

where is growing measurable,

To obtain the reverse implication we use standard test function and Lemma 2.5

where
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

Let us note that Theorems 3.2 and 3.7 of [3] show that under some restrictions on the following assertion is true.
For every function belongs to
and the reverse is also true, for any function
there exists "an extension"
such that

The following Theorem 4.6 gives an answer for the same map from polydisk to expanded disk

for functions from
We will develop ideas from [4] to get the following sharp embedding theorem for classes on expanded disk.
Theorem 4.6.
Let be a positive Borel measure on
Then

if and only if
Proof.
Obviously if then by putting

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

In what follows we will use notations of Proposition 2.1. Consider the Poisson integral of a function (see [2]).
Let We will show now as in [4] that

Indeed, this will be enough, since the operator is
operator we can apply the Marcinkiewicz interpolation theorem (see [16, Chapter 1]) to assert that

We have as in [4] for

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

Theorem 4.6 is proved.
Theorem 4.7.
Let Then

where

where

where
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


and estimate


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

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]).
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 function is done again from diagonal
Let
and

where is a constant of Bergman representation formula (see [1]).
Proposition 4.9.

Then

And reverse is also true:

Let and

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

Let Then

and the reverse is also true:

For any function with a finite quasinorm
such that condition (4.38) holds for
one has
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 It remains to apply Theorem A.

For the proof of we use Theorem 4.6 and get the result we need.
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.

Let then the following assertions are true: If
then

If then

Let If
then
Moreover the reverse is also true if condition (4.38) holds for
then
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 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 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.
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Acknowledgment
The authors sincerely thank Trieu Le for valuable discussions.
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Shamoyan, R.F., Mihić, O.R. Analytic Classes on Subframe and Expanded Disk and the Differential Operator in Polydisk.
J Inequal Appl 2009, 353801 (2009). https://doi.org/10.1155/2009/353801
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DOI: https://doi.org/10.1155/2009/353801
Keywords
- Holomorphic Function
- Unit Disk
- Bergman Space
- Carleson Measure
- Dyadic Cube