On Traces in Some Analytic Spaces in Bounded Strictly Pseudoconvex Domains

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.


Introduction and Formulation of Problem
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 [1][2][3][4] where similar results were obtained but only in simpler bounded domains in higher dimension.We remark that in this note for the first time in the literature we consider this known problem related with trace estimates in spaces of analytic functions in more general pseudoconvex domains in C  , namely, in strictly pseudoconvex domains with smooth boundary.The next section contains required preliminaries on analysis on bounded strictly pseudoconvex domains with smooth boundary.Our new sharp results are contained in the last section of this note.Related estimates for Bergman type projections will be also provided.All our main results in context of unit ball can be seen in [2][3][4].All preliminary assertions of this paper have their direct analogues in context of unit ball and this can be seen in [2][3][4] and references therein.It is known that the geometry of pseudoconvex domains is more complicated and extra arguments were needed to get technical lemmas which are used to prove main results of this paper.These subtle lemmas can be seen in particular in recent papers [5,6].Now we will shortly present the history of the diagonal map (or traces) problem.After the appearance of [7], various papers appeared where arguments which can be seen in [7] were extended, changed, and modified in various directions in one and higher dimension (see, e.g., [1,3,8,9] and also various references therein).In particular in mentioned papers various new sharp results on traces for analytic function spaces in higher dimension (unit polyball) were obtained.New results for large scales of analytic   type spaces in polyball were proved (see [4]).Later several new sharp results for harmonic functions of several variables in the unit ball and upper half-plane of Euclidean space were also obtained (see, e.g., [1] and references therein).For the first time in the literature, these types of problems connected with diagonal map in analytic spaces appeared before in [7].In [7], this problem was formulated and certain concrete cases connected with spaces of analytic functions in the unit disk were considered.Some interesting applications of diagonal map can be seen in [8,10] where other problems around this topic can be found also.The goal of this note is to develop further some ideas from our recent mentioned papers and present new sharp theorems in strictly pseudoconvex domains with smooth boundary.
Extension problems were studied mainly by two different methods.The one is the extension using integral formula in the case where  is bounded pseudoconvex domains with a support function (domains with smooth boundary).The other is the  2 extension using the Hilbert space theory in the case when  is general bounded pseudoconvex domain (see [11][12][13][14]).
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 [2][3][4]).
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 Tr () = (, . . ., ),  ∈ , if  ∈  ⊂ (  ) for certain quasinormed space  on  products of  domains   , where (  ) is a space of analytic functions in products domains   .
Note these type maps were previously considered by various authors in particular cases when Ω = D (unit disc), when Ω = B (unit ball).Applications of this map to various problems in function theory are also known (see, e.g., [2,4,10] and references therein).
Note in addition that we use heavily the same machinery which was recently developed in [5,6].
The trace problem in particular is in short the following.Let  ∈ (  ),   =  × ⋅ ⋅ ⋅ ×  and  satisfies certain growth condition  ∈ ,  ⊂ (  ) and then get as much information as possible about growth of (, . . ., ),  ∈ , where  can be certain fixed functional class of analytic functions ( 1 , . . .,   ),   ∈ ,  = 1, . . .,  (analytic by each variable).We also will look at the same time at estimates of various multifunctional operators and expressions closely related with restriction map.
The technique we use is based also on a work of Beatrous (see [12]) and Ortega-Fabrega (see [15]) and some information from [5,6] on pseudoconvex domains (namely, some subtle estimates from very recent papers [5,6]) will be also used.Note that various similar extension theorems were previously studied by many authors (Henkin, Adachi, and Cumenge); see [11][12][13]15] and various references therein.
Actually in this paper we continue (partially) the investigation of Jimbo and Sakai (see [16]) related to function spaces on products of pseudoconvex domains in C  .
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: where  ∈ (  ) and   > 0,  = 1, . . ., .
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  is a unit disk).Note now our goal in this paper to study only particular case of general  , , classes and integral operators (Bergman type) on them.Note also that Bergman type projection from various points of view in pseudoconvex domains was studied before via various authors (see, e.g., [12,15,[17][18][19] and references therein).
The study of traces of general  , , spaces (including limit case) is a separate problem which will be addressed by authors in a separate paper.
The study of analytic spaces in products of pseudoconvex domains was started probably in [16,20].This paper is heavily based on arguments that appeared in proofs of results from [2,4].Using them, we here extend various assertions which can be found in unit ball in [2,4] to the case of general strictly pseudoconvex bounded domains  with smooth boundary.
Throughout the paper , sometimes with indexes, stands for various positive constants which can be different even in a chain of inequalities and are independent of the discussed functions or variables.
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  < .

Preliminaries and Formulations of Main Theorems
In this section, we collect preliminaries and formulations of all main results of this paper.We define Bergman spaces on polypseudoconvex domains as where  is the Lebesgue measure on the  domain, 0 <  < ∞, and   > −1,  = 1, . . ., .These are Banach spaces for 1 <  < ∞ and complete metric spaces for 0 <  ≤ 1.We put   () =   ()(),  > −1.
Let now  be a  ∞ -bounded strongly pseudoconvex domain with defining function .We need some results for our proofs.We let (, ) be the associated Levi polynomial (see [21]).Consider It follows from Taylor's formula and the strict plurisubharmonicity of  that there are positive constants for ,  ∈   and | − | ≤  and g(, ) = (, ) for  ∈ .Also we have where N is the complex normal vector field of type (1, 0) defined by N = ∑  =1 (/  )(/  ).
such that Lemma B (see [21]).For each  > −1, there is a smooth form In this paper we deal with the following kernel  +1+ (, ) of  +  + 1 type which is the reproducing Bergman kernel for weighted Bergman spaces in pseudoconvex domains with smooth boundary (see [12,22]).
Proof.If  ∈    (), then for large enough  we have  ∈  1   ().The proof of these facts follows from the well-known proof in the unit disk case.Then, use Lemma C.
The study of Φ =  ,  operator (expanded Bergman projection) is of special interest.This operator was considered by many authors before in various situations (polydisk, unit ball, and spaces of harmonic functions in R +1 + and B) and used in relation to traces problems (see [2,4] and references therein).
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    spaces.Our results again are heavily based on lemmas from [5,6].
Let also The second part of the following theorem generalizes partially Theorem 8.
Remark 16.Exact values of  0 and  0 in Theorem 15 can be calculated by readers.We refer the reader to unit ball case where details can be seen (see [2][3][4]).
Remark 17.The proof of Theorem 15 will be omitted by us.In the unit ball it can be seen in [2,3,16].Moreover, the proof is based on same ideas as the proof of Theorems 1-14 with small modifications.

Proofs of Theorems 1-15 and Final Comment
In this section we provide proofs of all our main assertions which we formulated in this paper in the previous section.
From (34) we have using Propositions 3-4 finally where where   = ( + 1) +   ,  = 1, . . ., .We used at the last step the fact that {  (  , )},  = (1/2)(1+) family is a finite covering of , and so A very careful analysis of the proof of Theorem 1 shows that we replaced properties of (  , ) -lattices of the unit ball of C  (see [2,4] and references therein) with similar properties of -lattice invented recently in strictly pseudoconvex domains in important papers [5,6].This direction of arguments (replacement of -lattices) can be applied practically to all assertions from [2,4]; we partially formulated some parallel assertions above.In view of the mentioned similarities, we give now complete proofs of Theorems 12 and 14 leaving proofs of other assertions to readers and referring to arguments from [2,4].

Journal of Function Spaces
Remark 18.Note the assertion we just proved is in a little bit general form and can be found in Theorem 15.We omit details referring the readers to [2,4] where unit ball case was considered.
Remark 19.Note it is easy to see the same arguments are valid if we replace in formulations of our theorems kernels  / by |  | 1/ to get the same results.
The next lemma, Lemma D, as in unit ball case is playing the crucial role in the proof of Theorem 14 ( ≤ 1 case).We however show Theorem 14 using additional assumptions on   kernel.These assumptions (which are valid in the unit ball also) can be dropped using Lemma C and this is also completely similar to the proof of the unit ball case.
In unit disk this lemma can be seen in [8].In unit ball this lemma can be seen in [4].This lemma is crucial for the proof of trace theorems in unit disk and unit ball.Moreover, for  = 1 case in pseudoconvex domains, we refer the reader to [12].
Proof of Theorem 14.We start with the proof of the first part of our theorem.As in proof of the previous theorem, for every positive large enough   ,  = 1, . . ., , we have (, . . ., ) = (), where  ( by Corollary 3.9 from [12] for large enough β and the first part of Theorem 14 is proved. We turn now to the proof of the second part of Theorem 14, the case of   ⃗  spaces for  ≤ 1.The proof follows directly from Theorem 1 (the proof was provided above) and from the third part of Theorem 15.We turn to the proof of the third part of Theorem 15 completing the proof at the same time again using properties of -lattices of Kobayashi ball and we have the following inequalities.