Pseudodifferential operators with generalized symbols and regularity theory

Date

2005-10-21

Authors

Garetto, Claudia
Gramchev, Todor
Oberguggenberger, Michael

Journal Title

Journal ISSN

Volume Title

Publisher

Texas State University-San Marcos, Department of Mathematics

Abstract

We study pseudodifferential operators with amplitudes ɑε(x, ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 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 pseudo-differential 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.

Description

Keywords

Pseudodifferential operators, Small parameter, Slow scale net, Algebras of generalized functions

Citation

Garetto, C., Gramchev, T., & Oberguggenberger, M. (2005). Pseudodifferential operators with generalized symbols and regularity theory. <i>Electronic Journal of Differential Equations, 2005</i>(116), pp. 1-43.

Rights

Attribution 4.0 International

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