General coorbit space theory for quasi-Banach spaces and inhomogeneous function spaces with variable smoothness and integrability

In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gr{\"o}chenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov-Lizorkin-Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.