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Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities
journal contributionposted on 2015-04-17, 14:09 authored 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.
- Mathematical Sciences
Published inProceedings of the Edinburgh Mathematical Society
Pages603 - 629
CitationGARETTO, 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.
PublisherCambridge University Press / © Edinburgh Mathematical Society
- SMUR (Submitted Manuscript Under Review)
Publisher statementThis 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/
NotesThis 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