  We give characterizations for homogeneous and inhomogeneous
Besov-Lizorkin-Triebel spaces in terms of continuous local means for the full
range of parameters. In particular, we prove characterizations in terms of
Lusin functions and spaces involving the Peetre maximal function to apply the
classical coorbit space theory due to Feichtinger and Gr\"ochenig. This results
in atomic decompositions and wavelet bases for homogeneous spaces. In
particular we give sufficient conditions for suitable wavelets in terms of
moment, decay and smoothness conditions.
