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

TL;DR

This paper develops a comprehensive theory of global wave-front sets for various function spaces, including Banach, Fréchet, and modulation spaces, and studies their behavior under pseudo-differential operators.

Contribution

It introduces a unified framework for wave-front sets in different spaces and extends classical properties like micro locality and microellipticity to these settings.

Findings

Wave-front sets are defined for Banach, Fréchet, and modulation spaces.

Standard properties of wave-front sets extend to these new definitions.

Pseudo-differential operators act continuously and preserve micro locality and microellipticity.

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.