Pseudodifferential operators on -modulation spaces

We study expansions of pseudodifferential operators from the Hörmander class in a special family of functions called brushlets. We prove that such operators have a sparse representation in a brushlet system. Using this sparsity, we show that a pseudodifferential operator extends to a bounded operator between α-modulation spaces. These spaces were introduced by Gröbner in [15]. They are, in some sense, intermediate spaces between the classical Besov and Modulation spaces.