  In this paper we show that the Fourier transform induces an isomorphism
between the coorbit spaces defined by Feichtinger and Gr\"ochenig of the mixed,
weighted Lebesgue spaces $L_{v}^{p,q}$ with respect to the quasi-regular
representation of a semi-direct product $\mathbb{R}^{d}\rtimes H$ with suitably
chosen dilation group $H$, and certain decomposition spaces
$\mathcal{D}\left(\mathcal{Q},L^{p},\ell_{u}^{q}\right)$ (essentially as
introduced by Feichtinger and Gr\"obner), where the localized ,,parts`` of a
function are measured in the $\mathcal{F}L^{p}$-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.
