Electron. J. Diff. Eqns., Vol. 2005(2005), No. 116, pp. 1-43.

Pseudodifferential operators with generalized symbols and regularity theory

Claudia Garetto, Todor Gramchev, Michael Oberguggenberger

Abstract:
We study pseudodifferential operators with amplitudes $a_\varepsilon (x,\xi)$ depending on a singular parameter $\varepsilon \to 0$ with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for $\varepsilon \to 0$, refined versions of estimates for classical pseudodifferential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudodifferential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations.

Submitted June 13, 2005. Published October 21, 2005.
Math Subject Classifications: 35S50, 35S30, 46F10, 46F30, 35D10.
Key Words: Pseudodifferential operators; small parameter; slow scale net; algebras of generalized functions.

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Claudia Garetto
Institut für Technische Mathematik
Geometrie und Bauinformatik
Universität Innsbruck
A - 6020 Innsbruck, Austria
email: claudia@mat1.uibk.ac.at
Todor Gramchev
Dipartimento di Matematica e Informatica
Universitá di Cagliari, I - 09124 Cagliari, Italia
email: todor@unica.it
Michael Oberguggenberger
Institut für Technische Mathematik
Geometrie und Bauinformatik
Universität Innsbruck
A - 6020 Innsbruck, Austria
email: michael@mat1.uibk.ac.at

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