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

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 $(\alpha)$ 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 $\lambda(A-\lambda)^{-1}$ for $\Re(\lambda) \geq 0$. 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.