Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition

TL;DR

This paper extends the theory of coorbit spaces linked to integrable group representations and quasi-Banach function spaces, broadening applicability and simplifying previous results, including molecular decompositions and dual frames.

Contribution

It generalizes coorbit space theory to arbitrary locally compact groups and weaker integrability conditions, and establishes molecular dual frames and Riesz sequences for quasi-Banach spaces.

Findings

Extended coorbit theory to nonunimodular groups.

Proved existence of molecular dual frames and Riesz sequences.

Applied weaker integrability conditions for analyzing vectors.

Abstract

This paper provides a self-contained exposition of coorbit spaces associated to integrable group representations and quasi-Banach function spaces, and at the same time extends and simplifies previous work. The main results provide an extension of the theory in [Studia Math., 180(3):237-253, 2007] from groups admitting a compact, conjugation-invariant unit neighborhood to arbitrary (possibly nonunimodular) locally compact groups. In addition, the present paper establishes the existence of molecular dual frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the full scale of quasi-Banach function spaces. The theory is developed for possibly projective and reducible unitary representations in order to be easily applicable to well-studied function spaces not satisfying the classical assumptions of coorbit theory. Compared to the existing literature on quasi-Banach coorbit spaces, all our results apply under significantly weaker integrability conditions on the analyzing vectors, which allows for obtaining sharp results in concrete settings