Analytic Classes on Subframe and Expanded Disk and the Differential Operator in Polydisk

Let and be the -dimensional space of complex coordinates. We denote the unit polydisk by

(11)

and the distinguished boundary of by

(12)

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

(13)

and the subframe by

(14)

The Hardy spaces, denoted by are defined by

(15)

where

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

(16)

For the Bergman class on expanded disk is defined by

(17)

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

(18)

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.

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:

(21)

where and are any positive Borel measures on and such that

(22)

where (b) Let Then

(23)

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

(24)

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

(25)

So the lemma will be proved if Let

(26)

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

The last estimate follows from inclusion

(27)

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

(28)

It remains to note that we put above

1. (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

(210)

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

(211)

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

(212)

Corollary 2.6.

Let Then

(213)

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

Theorem 2 A (see [3]).

Let Then

(214)

Let Then

(215)

Theorem 2 B (see [3]).

Let Then

(216)

Let Then

(217)

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

(31)

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:

(32)

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:

(33)

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

(34)

by Bergman representation formula. Obviously , as

(35)

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

(36)

where

We will need the following assertion. Let

Then let

(37)

where We assert that belongs to

Indeed using Hölder's inequality we get

(38)

where We used above the estimate

(39)

Returning to estimate for we have by Hölder's inequality

(310)

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

(311)

Indeed by Hölder's inequality we have

(312)

Hence calculating integrals we finally have

We used the estimate

(313)

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

(314)

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,

(315)

Let

(316)

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

(317)

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

(318)

We have for such a function

(319)

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

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]):

(41)

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:

(42)

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:

(43)

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

(44)

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

(45)

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

(46)

Note further since

(47)

We have

(48)

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

(49)

for any Hence and for by Lemma 2.3

(410)

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

(411)

We used estimate

(412)

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

(413)

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

(414)

then

(415)

and conversely if

(416)

holds for some where then

(417)

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:

(418)

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

(419)

where is growing measurable,

(420)

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

(421)

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

(422)

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

(423)

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

(424)

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

(425)

if and only if

Proof.

Obviously if then by putting

(426)

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

Let

Let Let also and where

(427)

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

(428)

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

(429)

We have as in [4] for

(430)

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

(431)

Theorem 4.6 is proved.

Theorem 4.7.

Let Then

(432)

where

(433)

where

(434)

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

(435)

and estimate

(436)

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

(437)

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

(438)

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

Proposition 4.9.

Then

(439)

And reverse is also true:

Let and

(440)

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

Let Then

(441)

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

(442)

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

(443)

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.