On the solvability of systems of pseudodifferential operators
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).
Keywords
Cite
@article{arxiv.0801.4043,
title = {On the solvability of systems of pseudodifferential operators},
author = {Nils Dencker},
journal= {arXiv preprint arXiv:0801.4043},
year = {2010}
}
Comments
Changed Definition 2.5 and corrected the proof of Proposition 2.12. Rewrote Section 2, corrected errors and misprints. Corrected some references and the formulation of Theorem 2.7 and Remark 6.1. The paper has 40 pages