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Abstract

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

Keywords

tempered distributionsglobal wave front setsmicrolocal analysisphase spaceanisotropypropagation of singularitiesevolution equations

MSC classification

Primary: 46F05: Topological linear spaces of test functions, distributions and ultradistributions 46F12: Integral transforms in distribution spaces 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE 47G30: Pseudodifferential operators 35S05: Pseudodifferential operators 35A18: Wave front sets 81S30: Phase-space methods including Wigner distributions, etc. 58J47: Propagation of singularities; initial value problems 47D06: One-parameter semigroups and linear evolution equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 25
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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