Bergman-type operators in tubular domains over symmetric cones

Abstract

We study the boundedness properties of Rudin-Forelli-type operators associated to tubular domains over symmetric cones. As an application, we give a characterization of the topological dual space of the weighted Bergman space A p,q ν .

A Note on a Distance Function in Bergman Type Analytic Function Spaces of Several Variables

“…The following result can be found in [18] (section 4). We remark this result is a particular case of a more general assertion for analytic mixed norm A p,q ν classes (see, for example, [18]), which (after some analysis of our proof below) means that our main result admits also some extensions, even to mixed norm spaces which we defined above.…”

“…For formulation of our results we will need various standard definitions from the theory of tubular domains over symmetric cones (see, for example, [4,[17][18][19]). …”

“…For various properties of such functions we refer the reader to [4]. Replacing above A by L we will get as usual the corresponding larger space of all measurable functions in tube over symmetric cone with the same quazinorm (see [17,18]). …”

“…As previously in case of analytic functions in unit disk, polydisk, unit ball, and upperhalfspace C + and in case of spaces of harmonic functions in Euclidean space (see, for example, [1][2][3][5][6][7]) the role of the Bergman representation formula is crucial in these issues and our proof is heavily based on it. A variant of Bergman representation formula is available also in Bergman-type analytic function spaces in tubular domains over symmetric cones and this known fact (see [4,[17][18][19]), which is crucial also in various problems in analytic function spaces in tubular domains (see [4] and various references there) is also used in our proof below.…”

Schatten class Toeplitz operators on weighted Bergman spaces of tube domains over symmetric cones

“…The proof of the other parts essentially uses the properties of Bergman balls and the δ-lattices. We also refer to [15,17] for this type of results.…”