Almost Diagonalization of τ-Pseudodifferential Operators with Symbols in Wiener Amalgam and Modulation Spaces

In this paper we focus on the almost-diagonalization properties of tau-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. Such spaces are nowadays called modulation spaces as well. A particular example is provided by the Sjostrand class, for which Grochenig (Rev Mat Iberoam 22(2):703-724, 2006) exhibits the almost diagonalization of Weyl operators. We shall show that such result can be extended to any tau-pseudodifferential operator, for tau is an element of[0,1]. This is not surprising, since the mapping that goes from a Weyl symbol to a tau-symbol is bounded in the Sjostrand class. What is new and quite striking is the almost diagonalization for tau-operators with symbols in weighted Wiener amalgam spaces. In this case the diagonalization depends on the parameter tau. In particular, we have an almost diagonalization for tau is an element of(0,1) whereas the cases tau=0 or tau=1 yield only to weaker results. As a consequence, we infer boundedness, algebra and Wiener properties for tau-pseudodifferential operators on Wiener amalgam and modulation spaces.