^{1}

^{2}

^{1}

^{2}

New sharp estimates of traces of Bergman type spaces of analytic functions in bounded strictly pseudoconvex domains are obtained. These are, as far as we know, the first results of this type which are valid for any bounded strictly pseudoconvex domains with smooth boundary.

In this note we obtain new sharp estimates for traces in Bergman type spaces of analytic spaces in strictly pseudoconvex domains with smooth boundary. This line of investigation can be considered as a continuation of our previous papers on traces in analytic function spaces [

Now we will shortly present the history of the diagonal map (or traces) problem. After the appearance of [

Some interesting applications of diagonal map can be seen in [

Extension problems were studied mainly by two different methods. The one is the extension using integral formula in the case where

For formulation of our results, we will need various standard definitions from the theory of strictly pseudoconvex domains with smooth boundary. In this and next section, we mention some vital facts which will be heavily used in proofs of our assertions (see, e.g., also for parallel assertion in other domains [

Let

Let (see [

For

Let

One of the intentions of this paper is to consider new trace problem and trace map for this case of bounded strictly pseudoconvex domains with smooth boundary. It is a map

Note these type maps were previously considered by various authors in particular cases when

Note in addition that we use heavily the same machinery which was recently developed in [

The trace problem in particular is in short the following. Let

The technique we use is based also on a work of Beatrous (see [

Actually in this paper we continue (partially) the investigation of Jimbo and Sakai (see [

To define new Bergman type analytic spaces on products of pseudoconvex domains we have to replace one integral by multiple integrals in spaces above. For example, the mentioned quasinorms will take this form:

We formulate in the next section some new results related to restriction maps in products of pseudoconvex domains and multifunctional spaces generalizing previous estimates for polydisk (when

The study of traces of general

The study of analytic spaces in products of pseudoconvex domains was started probably in [

Throughout the paper

The notation

In this section, we collect preliminaries and formulations of all main results of this paper.

We define Bergman spaces on polypseudoconvex domains as

Let

Let

The proof of this theorem is based only on Propositions

Let

Let

Let

Let now

It follows from Taylor’s formula and the strict plurisubharmonicity of

Let

for any

there is a nonvanishing

For each

for

Let

In this paper we deal with the following kernel

We need also estimates for Bergman type kernel, the so-called Forelli-Rudin type estimates. The following assertion is valid if we change the index of kernel

Let

Estimate (

Let

Let us further define Bergman type integral operators. If

Let

Let

If

We now also note that the following is true. Let

The study of

We provide some new estimates here for expanded Bergman projection based on our fully previous work in unit ball extending known estimates for ordinary Bergman projection in strictly pseudoconvex domains in weighted Bergman

Let

It will be interesting for reader to compare this result with Theorem 4.1 from [

A variant of Theorem

From Proposition

Let

Note that a trace theorem can be extended to some mixed norm classes defined like this

We formulate now new two trace theorems concerning

Let also

Let

Trace

Trace

The second part of the following theorem generalizes partially Theorem

(1) Let

(2) Let

(3) Let

Exact values of

Exact values of

The proof of Theorem

In this section we provide proofs of all our main assertions which we formulated in this paper in the previous section.

We have the following chain of estimates using properties of Kobayashi balls and Propositions

From (

A very careful analysis of the proof of Theorem

Remark

Note that for

An application of Fubini’s theorem and the same estimate we just used above lead finally to the estimate that is

The theorem is proved.

Note the assertion we just proved is in a little bit general form and can be found in Theorem

Note it is easy to see the same arguments are valid if we replace in formulations of our theorems kernels

The next lemma, Lemma D, as in unit ball case is playing the crucial role in the proof of Theorem

Let

In unit disk this lemma can be seen in [

We start with the proof of the first part of our theorem. As in proof of the previous theorem, for every positive large enough

Note first that it is obvious

to show the reverse we get by Hölder’s inequality from (

We turn now to the proof of the second part of Theorem

The proof follows directly from Theorem

Here we need

These facts and Propositions

We put additional condition on Bergman kernel below, but with the help of Lemma D it can be removed. Note similar arguments we used in the case of unit ball. We have now using (

The proof of Theorem

Some results of this paper can be extended to the so-called analytic Herz type spaces in product domains. To define the Herz space based on Kobayashi balls, we remind the reader that there exists a family of Kobayashi balls

It will be interesting to extend our results to

Some results of this paper can be also obtained by similar technique in bounded symmetric domains and bounded minimal homogeneous domains in higher dimension.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Olivera R. Mihić was supported by MNTR Serbia, Project 174017.