New sharp estimate concerning distance function in certain Bergman-type spaces of analytic functions on tube domains over symmetric cones is obtained. This is the first result of this type for tube domains over symmetric cones. New similar results in analytic mixed norm spaces on products of tube domains over symmetric cones will also be provided.

DOI: 10.17516/1997DOI: 10.17516/ -1397DOI: 10.17516/ -2016 In this note we give new estimates for traces of new mixed norm Hardy type spaces in tubular domains over symmetric cones and in bounded strictly pseudoconvex domains. We generalize a well-known one dimensional result concerning traces of Hardy spaces obtained previously in the unit disk by various authors. This trace problem in Hardy spaces(diagonal map problem) in the unit disk and in the unit polydisk was considered by many authors during last several decades. In polydisk some estimates related with this problem can be seen, for example, in [1,2]. In polydisk, but in particular values of parameters in [3] and in some papers from list of references of [1,2]. A rather long history related with these type estimates of traces of Hardy spaces can be read in [4] and in [2] also. In this paper we extend a known crucial estimate related with this problem to more general and complicated cases of tubular domains over symmetric cones and pseudoconvex domains with smooth boundary putting a natural condition on Bergman kernel of these domains. This paper can be considered also as continuation of a long series of papers of first author on traces of function spaces (see, for example, [5][6][7][8] and various references there).In recent decades many papers appeared where various Hardy and other analytic spaces were studied from various points of views in higher dimension in various domains in C n . We refer for example to a series of papers of Krantz and coauthors (see [9][10][11] in particular) and also [12][13][14][15] in this direction. For some new interesting results on analytic spaces in tubular domains over symmetric cones we refer the reader to [16][17][18][19] and various references there also. We will heavily use nice techniques which was developed in these papers related with so-called lattices.We start hoverer with a result in the unit ball. Then we define new mixed norm Hardy type classes in tubular domains and pseudoconvex domains (see [1,2] for much simpler case of the unit polydisk). Then we provide a complete proof of our assertion in polyball and then provide assertions in bounded pseudoconvex domains with smooth boundary and in tubular domains over symmetric cones. In all cases proofs actually are the same. *

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We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball. Namely we prove that mΠ j=1 jjfj jjXj ≍ jjf1 : : : fmjj Ap for various (Xj) spaces of analytic functions in bounded pseudoconvex domains with smooth boundary where f; fj ; j = 1; : : : ;m are analytic functions and where Ap ; 0 < p < 1; > �����1 is a Bergman space. This in particular also extend in various directions a known theorem on atomic decomposition of Bergman Ap spaces.