R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

Robert Denk, Thomas Krainer

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (HT) and Pisier's property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of λ(A-λ)-1 in R for some R. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on R with operator valued coefficients.

Original languageEnglish (US)
Pages (from-to)319-342
Number of pages24
JournalManuscripta Mathematica
Volume124
Issue number3
DOIs
StatePublished - Nov 1 2007

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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