Coorbit spaces associated to integrably admissible dilation groups

This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any integrably admissible dilation group there exists a convienent space of smooth, admissible analyzing vectors that can be used to define a consistent coorbit space possessing all the essential properties that are known to hold in the setting of discrete series representations. In particular, the obtained coorbit spaces can be realized as Besov-type decomposition spaces by means of a Littlewood—Paley-type characterization. The classes of anisotropic Besov spaces associated to expansive matrices are shown to coincide precisely with the coorbit spaces induced by the integrably admissible one-parameter groups.