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Pseudodifferential operators with generalized symbols and regularity theory
journal contributionposted on 2015-04-17, 15:10 authored by Claudia Garetto, Todor Gramchev, Michael Oberguggenberger
We study pseudodifferential operators with amplitudes aε(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 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. © 2005 Texas State University - San Marcos.
C. Garetto was supported by INDAM–GNAMPA, Italy. T. Gramchev was supported by INDAM–GNAMPA, Italy and by grant PST.CLG.979347 from NATO. M. Oberguggenberger was supported by project P14576-MAT from FWF, Austria.
- Mathematical Sciences
Published inElectronic Journal of Differential Equations
CitationGARETTO, C., GRAMCHEV, T. and OBERGUGGENBERGER, M., 2005. Pseudodifferential operators with generalized symbols and regularity theory. Electronic Journal of Differential Equations, 116, pp. 1-43.
Publisher© Department of Mathematics, Texas State University
- VoR (Version of Record)
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 article was published in the Electronic Journal of Differential Equations [© Texas State University]. It is also available at: http://ejde.math.txstate.edu/Volumes/2005/116/garetto.pdf