On the solvability of systems of pseudodifferential operators

TL;DR

This paper investigates the conditions under which systems of pseudodifferential operators are locally solvable, establishing an equivalence with a specific eigenvalue condition and providing a priori estimates without lower order term restrictions.

Contribution

It introduces a new solvability criterion (PSI) for principal type systems and proves local solvability with a derivative loss, without requiring conditions on lower order terms.

Findings

Solvability is equivalent to the eigenvalue condition (PSI).

Local solvability is achieved with a 3/2 derivative loss.

No conditions on lower order terms are necessary.

Abstract

The paper studies the solvability for square systems of pseudodifferential operators. We assume that the system is of principal type, i.e., the principal symbol vanishes of first order on the kernel. We shall also assume that the eigenvalues of the principal symbol close to zero have constant multiplicity. We prove that local solvability for the system is equivalent to condition (PSI) on the eigenvalues of the principal symbol. This condition rules out any sign changes from - to + of the imaginary part of the eigenvalue when going in the positive direction on the bicharacteristics of the real part. Thus we need no conditions on the lower order terms. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of 3/2 derivatives (compared with the elliptic case).