Solvability and limit bicharacteristics

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Abstract

We study the solvability of pseudodifferential operators which are not of principal type. The operator will have real principal symbol and we shall consider the limits of bicharacteristics at the set where the principal symbol vanishes of at least second order. The convergence shall be as smooth curves, then the limit bicharacteristic also is a smooth curve. We shall also need uniform bounds on the curvature of the characteristics at the bicharacteristics, but only along the tangents of a given Lagrangean manifold. This gives uniform bounds on the linearization of the normalized Hamilton flow on the tangent space of this manifold at the bicharacteristics. If the quotient of the imaginary part of the subprincipal symbol with the norm of the Hamilton vector field switches sign from − to + on the bicharacteristics and becomes unbounded as they converge to the limit, then the operator is not solvable at the limit bicharacteristic.

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Subject classification (UKÄ) – MANDATORY

  • Mathematical Analysis

Keywords

  • Limit bicharacteristic, Pseudodifferential operator, Solvability, Subprincipal symbol
Original languageEnglish
Pages (from-to)295-320
Number of pages26
JournalJournal of Pseudo-Differential Operators and Applications
Volume7
Issue number3
Publication statusPublished - 2016 Sep 1
Publication categoryResearch
Peer-reviewedYes