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

2006/07/28 by Denk, Robert, Krainer, Thomas
#35K40 #35K65 #35R20 #58J05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/0607735

Abstract

It is shown that an elliptic scattering operator A on a compact manifold with boundary with coefficients in the bounded operators 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 of the manifold avoids the right half-plane. This is deduced directly from a Seeley theorem, i.e. the resolvent is represented in terms of pseudodifferential operators with R-bounded symbols, thus showing by an iteration argument the R-boundedness of λ(A-λ)^-1 for \Re(λ) ≥ 0. 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^d with operator valued coefficients.

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