On Some Microlocal Properties of the Range of a Pseudo-Differential Operator of Principal Type

TL;DR

This paper investigates microlocal properties of the range of principal type pseudo-differential operators, extending Hörmander's results on differential operators to operators that are not locally solvable, using advanced microlocal analysis techniques.

Contribution

It provides microlocal analogues of Hörmander's inclusion results for the range of certain pseudo-differential operators of principal type that fail condition $( ext{ extPsi})$.

Findings

Established microlocal inclusion relations for the range of principal type pseudo-differential operators.

Extended Hörmander's results from differential operators to pseudo-differential operators.

Analyzed properties of operators that do not satisfy the condition $( ext{ extPsi})$.

Abstract

The purpose of this paper is to obtain microlocal analogues of results by L. H \"ormander about inclusion relations between the ranges of first order differential operators with coefficients in $C^\infty$ which fail to be locally solvable. Using similar techniques, we shall study the properties of the range of classical pseudo-differential operators of principal type which fail to satisfy condition $(\Psi)$.