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Title: Pseudodifferential operators with generalized symbols and regularity theory
Authors: Garetto, Claudia
Gramchev, Todor
Oberguggenberger, Michael
Keywords: Algebras of generalized functions
Pseudodifferential operators
Slow scale net
Small parameter
Issue Date: 2005
Publisher: © Department of Mathematics, Texas State University
Citation: GARETTO, C., GRAMCHEV, T. and OBERGUGGENBERGER, M., 2005. Pseudodifferential operators with generalized symbols and regularity theory. Electronic Journal of Differential Equations, 116, pp. 1-43.
Abstract: 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.
Description: This 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
Sponsor: 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.
Version: Published
URI: https://dspace.lboro.ac.uk/2134/17305
Publisher Link: http://ejde.math.txstate.edu
ISSN: 1072-6691
Appears in Collections:Published Articles (Maths)

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