m-Microlocal elliptic pseudodifferential operators acting on $L^p_{\rm loc}(\Omega)$

TL;DR

This paper investigates the extension properties of weighted pseudodifferential operators on local L^p spaces and explores microlocal analysis and singularity propagation in solutions to pseudodifferential equations.

Contribution

It introduces non-homogeneous microlocal properties and studies the extension of pseudodifferential operators on L^p_{loc} spaces, with applications to singularity propagation.

Findings

Characterization of minimal and maximal extensions of operators

Introduction of non-homogeneous microlocal properties

Examples illustrating theoretical results

Abstract

In the first part of the paper the authors study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fr\'echet space $L^p_{\rm loc}(\Omega)$. In the second one non homogeneous microlocal properties are introduced and propagation of Sobolev singularities for solutions to (pseudo)differential equations is given. For both the arguments actual examples are provided.