A symbolic calculus and a parametrix construction for pseudodifferential operators with non-smooth negative definite symbols
type: Item
instance of: scholarly article
title: A symbolic calculus and a parametrix construction for pseudodifferential operators with non-smooth negative definite symbols
zbMATH Open document ID: 1170.47032
author: Alexander Potrykus
published in: Revista Matemática Complutense
publication date: 2009-04-02T00:00:00Z
full work available at URL: https://eudml.org/doc/42340
review text: The author introduces negative definite functions and a symbolic calculus for non-smooth negative definite symbols. A subclass of these symbols is introduced with the additional property that they are asymptotically constant in the co-variable. This technique and the part of Fredholm theory used in this work can be found in [\textit{H.\,Kumano-go}, ``Pseudo-differential operators'' (Cambridge/MS--London: MIT Press) (1982; Zbl 0489.35003)] for classes of classical symbols. The minimum differentiability that a symbol needs to have with respect to \(x\) depends on the order of the symbol. The author constructs Feller semigroups with non-smooth symbols by using the Hille--Yosida theorem and the (previously developed) symbolic calculus. Feller semigroups are strongly continuous contraction semigroups on \(C_\infty(\mathbb R^n)\) that are positivity preserving. An easy way to identify compact pseudodifferential operators leads to the construction of a parametrix by means of Fredholm theory, without using the sharp Gårding inequality or Friedrichs symmetrization, which are also available within the non-smooth calculus at hand.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5538725
zbMATH Keywords: symbolic calculus
zbMATH Keywords: nonsmooth symbols
zbMATH Keywords: negative definite functions
zbMATH Keywords: Feller semigroups
MaRDI profile type: Publication
OpenAlex ID: W2121550643
DOI: 10.5209/REV_REMA.2009.V22.N1.16347
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description: scientific article; zbMATH DE number 5538725