New Atomic Decompositions of Bergman Spaces on Bounded Symmetric Domains
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New Atomic Decompositions of Bergman Spaces on Bounded Symmetric Domains Jens Gerlach Christensen1
· Gestur Ólafsson2
Received: 18 May 2020 / Accepted: 30 October 2020 © Mathematica Josephina, Inc. 2020
Abstract We provide a large family of atoms for Bergman spaces on irreducible bounded symmetric domains. The atomic decompositions are derived using the holomorphic discrete series representations for the domain, and the approach is inspired by recent advances in wavelet and coorbit theory. Our results vastly generalize previous work by Coifman and Rochberg. Their atoms correspond to translates of a constant function at a discrete subset of the automorphism group of the domain. In this paper we show that atoms can be obtained as translates of any holomorphic function with rapidly decreasing coefficients (including polynomials). This approach also settles the relation between atomic decompositions for the bounded and unbounded realizations of the domain. Keywords Bounded domains · Bergman spaces · Atomic decomposition Mathematics Subject Classification 32A36 · 32A50
The research was partially supported by NSF Grant DMS 1321794 during the MRC program Lie Group Representations, Discretization, and Gelfand Pairs. The research of G. Ólafsson was also partially supported by Simons Grant 586106.
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Jens Gerlach Christensen [email protected] http://www.math.colgate.edu/~jchristensen Gestur Ólafsson [email protected] http://www.math.lsu.edu/~olafsson
1
Department of Mathematics, Colgate University, Hamilton, USA
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Department of Mathematics, Louisiana State University, Baton Rouge, USA
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J. G. Christensen, G. Ólafsson
1 Introduction This paper is concerned with providing atomic decompositions of Bergman spaces on bounded symmetric domains. The results extend similar results by Coifman and Rochberg [14] carried out for Bergman spaces on the unbounded realization of the domain and on the unit ball. Coifman and Rochberg asked if their decompositions would hold for the bounded domains, and in this paper we give a positive answer to this question. Moreover, we rectify an issue occurring in higher rank spaces which was pointed out in a remark on p. 614 in [2] (see also Remark 4 in [16]). While an extension to the bounded domains was predicted by Faraut and Koranyi in the introduction of [16] our results provide a much larger class of atoms than have previously been discovered. The usual atomic decompositions of Bergman spaces arise from a discretization of the integral reproducing formula, and atoms can thus be regarded as samples of the Bergman kernel in one of the variables. It turns out that this result can be formulated in terms of the holomorphic discrete series representation, in which case the classical atoms correspond to letting a discrete subset of the group of isometries act on a constant function. This viewpoint is extended widely in this paper where we show that atoms can be obtained by translates of any polynomial, or more generally, any analytic function with rapidly decreasing coefficients.
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