Coorbits for projective representations with an application to Bergman spaces

Abstract

Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.