  Coorbit space theory is an abstract approach to function spaces and their
atomic decompositions. The original theory developed by Feichtinger and
Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of
locally compact groups. Their theory covers, in particular, homogeneous
Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the
recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces
cannot be covered by their group theoretical approach. Later it was recognized
by Fornasier and the first named author that one may replace coherent states
related to the group representation by more general abstract continuous frames.
In the first part of the present paper we significantly extend this abstract
generalized coorbit space theory to treat a wider variety of coorbit spaces. A
unified approach towards atomic decompositions and Banach frames with new
results for general coorbit spaces is presented. In the second part we apply
the abstract setting to a specific framework and study coorbits of what we call
Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel
spaces of various types of interest as coorbits. We obtain several old and new
wavelet characterizations based on precise smoothness, decay, and vanishing
moment assumptions of the respective wavelet. As main examples we obtain
results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal
spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed
smoothness, and even mixtures of the mentioned ones. Due to the generality of
our approach, there are many more examples of interest where the abstract
coorbit space theory is applicable.
