Local means, wavelet bases and wavelet isomorphisms in Besov-Morrey and Triebel-Lizorkin-Morrey spaces

We consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r-regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov-Triebel-Lizorkin spaces.