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\({\mathcal{R}}\) -boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

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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.

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Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Fach D 193, Universität Konstanz, 78457, Konstanz, Germany

    Robert Denk

  2. Mathematics and Statistics, Penn State Altoona, 3000 Ivyside Park, Altoona, PA, 16601, USA

    Thomas Krainer

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  1. Robert Denk
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  2. Thomas Krainer
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Correspondence to Thomas Krainer.

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Denk, R., Krainer, T. \({\mathcal{R}}\) -boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators. manuscripta math. 124, 319–342 (2007). https://doi.org/10.1007/s00229-007-0131-1

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  • DOI: https://doi.org/10.1007/s00229-007-0131-1

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