  Starting with an integrable unitary representation of a locally compact group
and its associated voice transform, coorbit theory describes the construction
and investigation of the so-called coorbit spaces. A coorbit space consists of
distributions of which the voice transform is contained in a given function
space. We construct coorbit spaces which elements have voice transforms in
weighted Lebesgue spaces. By applying this construction to the wavelet
transform and short-time Fourier transform, we obtain special cases of the
homogeneous Besov spaces and the modulation spaces, respectively. We also
discretize the coorbit spaces in terms of Banach frames, where the frame
vectors are contained in the orbit of some suitable element under the group
action. For the two examples of the wavelet transform and the short-time
Fourier transform, we derive sufficient conditions for a wavelet respectively a
Gabor window to be suitable to generate such a Banach frame in the respective
context.
