On the microlocal properties of the range of systems of principal type

TL;DR

This paper investigates microlocal conditions governing the inclusion relations of ranges for systems of principal type pseudodifferential operators, extending scalar case results to more complex systems with constant characteristics.

Contribution

It extends previous scalar results to systems of principal type, analyzing microlocal inclusion relations and solvability conditions for non-locally solvable systems.

Findings

Identifies microlocal conditions for range inclusion

Extends scalar case results to systems of operators

Analyzes properties of systems with constant characteristics

Abstract

The purpose of this paper is to study microlocal conditions for inclusion relations between the ranges of square systems of pseudodifferential operators which fail to be locally solvable. The work is an extension of earlier results for the scalar case in this direction, where 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 were obtained. We shall study the properties of the range of systems of principal type with constant characteristics for which condition (\Psi) is known to be equivalent to microlocal solvability.