Local wave-front sets of Banach and Fréchet types, and pseudo-differential operators (Q1942328)

The definition of the Hörmander wave front set \(\text{WF}u\) of a distribution \(u\) can be generalized in different ways by replacing \(C^\infty\) regularity with other prescribed regularities, for example Sobolev \(H^s\) regularity. Here the authors propose a very general definition, by considering Banach spaces \(B\), with \(S(\mathbb{R}^n)\subset B\subset S'(\mathbb{R}^n)\), satisfying suitable properties which allow microlocalization and corresponding definition of \(\text{WF}_Bu\). Relevant examples of such spaces are given by the weighted modulation spaces, see, for example, [\textit{K. Gröchenig}, Foundations of time-frequency analysis. Boston, MA: Birkhäuser (2001; Zbl 0966.42020)] for the definition. The action of general pseudo-differential operators on \(\text{WF}_Bu\) is studied.