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Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities

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journal contribution
posted on 17.04.2015 by Claudia Garetto, Gunther Hormann
We characterize microlocal regularity, in the script G sign ∞-sense, of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow-scale generalized symbols. Thus we obtain an alternative, yet equivalent, way of determining generalized wavefront sets that is analogous to the original definition of the wavefront set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo)differential equations, where we extend the general non-characteristic regularity result for distributional solutions and consider propagation of script G sign ∞-singularities for homogeneous first-order hyperbolic equations.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Proceedings of the Edinburgh Mathematical Society

Volume

48

Issue

3

Pages

603 - 629

Citation

GARETTO, C. and HORMANN, G., 2005. Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proceedings of the Edinburgh Mathematical Society (Series 2), 48 (3), pp. 603 - 629.

Publisher

Cambridge University Press / © Edinburgh Mathematical Society

Version

SMUR (Submitted Manuscript Under Review)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2005

Notes

This paper was published in the journal, Proceedings of the Edinburgh Mathematical Society [Cambridge University Press / © Edinburgh Mathematical Society]. The definitive version is available at: http://dx.doi.org/10.1017/S0013091504000148

ISSN

0013-0915

Language

en

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