Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II
The author discusses general properties of pseudodifferential operators and Toeplitz operators in terms of weighted modulation spaces (MSs). The MS theory is extended in such a way that it might simplify applications of MSs to other fields, especially harmonic analysis. Basic properties of MSs are presented. It is shown that there is a canonical homeomorphism between MSs with different weight functions. In particular, the spaces \(M_s^{p,q}\) and \(M_{s,t}^{p,q}\) are constructed in a natural way from the standard space \(M^{p,q}\). This leads to embedding results between MSs and Sobolev/Besov spaces. By considering trace operators and extension operators in the background of MSs and using adjoint properties of these operators, one obtains some improvement in certain cases. From the above results, one obtains continuity results in operator theory for pseudodifferential operators and Toeplitz operators by making use of continuity properties of the ambiguity function in the context of MSs and relying on the role of this function in the Weyl calculus of pseudodifferential operators.
scientific article; zbMATH DE number 2093690