The Solvability and Subellipticity of Systems of Pseudodifferential Operators

TL;DR

This paper investigates the conditions under which systems of pseudodifferential operators are locally solvable and subelliptic, focusing on quasi-symmetrizable systems and their properties when the principal symbol vanishes of finite order.

Contribution

It introduces the concept of quasi-symmetrizable systems, proves their local solvability, and analyzes their subellipticity under finite order vanishing of the principal symbol.

Findings

Quasi-symmetrizable systems are locally solvable.

Subellipticity is established for systems with finite order vanishing of the principal symbol.

Condition (PSI) is extended to non-constant characteristic systems.

Abstract

The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (PSI) on the eigenvalues, see arXiv:0801.4043. This is a condition on the sign changes of the imaginary part of the eigenvalue along the oriented bicharacteristics of the real part. In the generic case when the principal symbol does not have constant characteristics, condition (PSI) is not sufficient, not invariant and in general not well defined. Instead we study systems which are quasi-symmetrizable, these systems have natural invariance properties and are of principal type. We prove that quasi-symmetrizable systems are locally solvable. We also study the subellipticity of quasi-symmetrizable systems in the case when principal symbol vanishes of finite order along the bicharacteristics. In order to prove subellipticity, we assume that the principal symbol has the approximation property, which implies that there are no transversal bicharacteristics.