MICROLOCAL ANALYSIS OF GENERALIZED FUNCTIONS: PSEUDODIFFERENTIAL TECHNIQUES AND PROPAGATION OF SINGULARITIES
By Claudia Garetto, G. Hörmann. Proceedings of the Edinburgh Mathematical Society, 2004
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Abstract We characterize microlocal regularity, in the $\mathcal{G}^{\infty}$-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 $\mathcal{G}^{\infty}$-singularities for homogeneous first-order hyperbolic equations.
Subfields: Family Practice, Mathematical Physics, Philosophy
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Citations in the corpus
- Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory (2005, background): “These investigations have produced a theory of point values in the Colombeau algebra G(Ω) and results of invertibility and positivity in the ring of constant generalized functions C̃ [19, 38, 39] but also microlocal analysis in Colombeau algebras and regularity theory for generalized solutions to pa”
- Topological Structures in Colombeau Algebras: Investigation of the Duals of ${\cal G}_{{\rm c}}(\Omega)$, ${\cal G}(\Omega)$ and ${\cal G}_{{\cal S}}({\Bbb R}^n)$ (2005, result): “The results obtained in this paper and in particular the chains of inclusions of Colombeau algebras and their duals set up a systematic framework where the regularity theory for pseudodifferential operators with generalized symbols [4, 8 , 7, 10, 11] can be settled and developed.”