Global wave-front sets of Banach, Fr{\'e}chet and Modulation space types, and pseudo-differential operators

Abstract

We introduce global wave-front sets $\operatorname{WF}_{{\mathcal B}} (f)$, $f\in {\mathscr S}^\prime(\textbf{R}^d)$, with respect to suitable Banach or Fr\'echet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(\omega,\mathscr B)$, where $\omega$ is an appropriate weight function and $\mathscr B$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to $\operatorname{WF}_{{\mathcal B}} (f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally defined pseudo-differential operators $\operatorname{Op}_t(a)$, acting continuously on the involved spaces.