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Spectral multiplier theorems via \(H^\infty \) calculus and R-bounds

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We prove spectral multiplier theorems for Hörmander classes \(\mathcal {H}^\alpha _p\) for 0-sectorial operators A on Banach spaces assuming a bounded \(H^\infty (\Sigma _\sigma )\) calculus for some \(\sigma \in (0,\pi )\) and norm and certain R-bounds on one of the following families of operators: the semigroup \(e^{-zA}\) on \(\mathbb {C}_+,\) the wave operators \(e^{isA}\) for \(s \in \mathbb {R},\) the resolvent \((\lambda - A)^{-1}\) on \(\mathbb {C}\backslash \mathbb {R},\) the imaginary powers \(A^{it}\) for \(t \in \mathbb {R}\) or the Bochner–Riesz means \((1 - A/u)_+^\alpha \) for \(u > 0.\) In contrast to the existing literature we neither assume that A operates on an \(L^p\) scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) \(\mathcal {H}^\alpha _1\) calculus in terms of R-boundedness of Bochner–Riesz means.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Blaise Pascal (UMR 6620), Université Clermont Auvergne, 63000, Clermont-Ferrand, France

    Christoph Kriegler

  2. Fakultät für Mathematik, Institut für Analysis, Karlsruher Institut für Technologie, Englerstrasse 2, 76131, Karlsruhe, Germany

    Lutz Weis

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  1. Christoph Kriegler
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  2. Lutz Weis
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Correspondence to Christoph Kriegler.

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The first named author acknowledges financial support from the Franco-German University (DFH-UFA) and the Karlsruhe House of Young Scientists (KHYS). The second named author acknowledges support from CRC 1173, DFG (Deutsche Forschungsgemeinschaft).

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Kriegler, C., Weis, L. Spectral multiplier theorems via \(H^\infty \) calculus and R-bounds. Math. Z. 289, 405–444 (2018). https://doi.org/10.1007/s00209-017-1957-1

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