Examples of Coorbit Spaces for Dual Pairs

TL;DR

This paper extends coorbit space theory to include new examples and non-integrable representations, broadening its applicability in characterizing various function spaces.

Contribution

It generalizes coorbit space theory to cover new Banach spaces and non-integrable representations, expanding its scope beyond previous limitations.

Findings

Provided new examples of coorbit spaces for dual pairs.

Achieved atomic decompositions for non-integrable representations.

Extended the characterization of function spaces like Bergman spaces.

Abstract

In this paper we summarize and give examples of a generalization of the coorbit space theory initiated in the 1980's by H.G. Feichtinger and K.H. Gr\"ochenig. Coorbit theory has been a powerful tool in characterizing Banach spaces of distributions with the use of integrable representations of locally compact groups. Examples are a wavelet characterization of the Besov spaces and a characterization of some Bergman spaces by the discrete series representation of $\mathrm{SL}_2(\mathbb{R})$. We present examples of Banach spaces which could not be covered by the previous theory, and we also provide atomic decompositions for an example related to a non-integrable representation.