$${\mathcal{R}}$$ -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 operator valued coefficients in the morphisms of a bundle of Banach spaces of class (\({\mathcal{HT}}\)) 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 \({\mathcal{R}}\) -bounded symbols, yielding by an iteration argument the \({\mathcal{R}}\) -boundedness of λ(A−λ)−1 in \(\Re(\lambda) \geq \gamma\) for some \(\gamma \in {\mathbb{R}}\) . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with \({\mathcal{R}}\) -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 \({\mathbb{R}}^{d}\) with operator valued coefficients.