Wavelet coorbit spaces viewed as decomposition spaces

In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces Lvp,q with respect to the quasi-regular representation of a semi-direct product Rd⋊HRd⋊H with suitably chosen dilation group H  , and certain decomposition spaces D(Q,Lp,ℓuq) (essentially as introduced by Feichtinger and Gröbner) where the localized “parts” of a function are measured in the FLpFLp-norm.This equivalence is useful in several ways: It provides access to a Fourier-analytic understanding of wavelet coorbit spaces, and it allows to discuss coorbit spaces associated to different dilation groups in a common framework. As an illustration of these points, we include a short discussion of dilation invariance properties of coorbit spaces associated to different types of dilation groups.