Coorbit spaces for dual pairs

In this paper we present an abstract framework for construction of Banach spaces of distributions from group representations. This extends the theory of coorbit spaces initiated by H.G. Feichtinger and K. Gröchenig in the 1980s. The coorbit theory sets up a correspondence between spaces of distributions and reproducing kernel Banach spaces. The original theory required that the initial representation was irreducible, unitary and integrable. As a consequence not all Bergman spaces could be described as coorbits. Our approach relies on duality arguments, which are often verifiable in cases where integrability fails. Moreover it does not require the representation to be irreducible or even come from a unitary representation on a Hilbert space. This enables us to account for the full Banach-scale of Bergman spaces on the unit disk for which we also provide atomic decompositions. Replacing the integrability criteria with duality also has the advantage that the reproducing kernel need not provide a continuous projection from a larger Banach function space. We finish the article with a wavelet characterization of Besov spaces on the forward light cone.