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This paper provides a self-contained exposition of coorbit spaces associatedto integrable group representations and quasi-Banach function spaces, and atthe same time extends and simplifies previous work. The main results provide anextension of the theory in [Studia Math., 180(3):237-253, 2007] from groupsadmitting a compact, conjugation-invariant unit neighborhood to arbitrary(possibly nonunimodular) locally compact groups. In addition, the present paperestablishes 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 functionspaces. The theory is developed for possibly projective and reducible unitaryrepresentations in order to be easily applicable to well-studied functionspaces not satisfying the classical assumptions of coorbit theory. Compared tothe existing literature on quasi-Banach coorbit spaces, all our results applyunder significantly weaker integrability conditions on the analyzing vectors,which allows for obtaining sharp results in concrete settings

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