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This paper provides a self-contained exposition of coorbit spaces associatedwith integrable group representations and quasi-Banach function spaces. Itextends the theory in [Studia Math., 180(3):237-253, 2007] to locally compactgroups that do not necessarily possess a compact, conjugation-invariant unitneighborhood. Furthermore, the present paper establishes the existence of dualmolecules of frames and Riesz sequences as in [J. Funct. Anal., 280(10):56,2021] for the setting of quasi-Banach spaces. To ensure the directapplicability to various well-studied examples, the theory is developed forpossibly projective and reducible unitary representations.

Coorbit spaces and dual molecules: The quasi-Banach case