Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type

  • Alessio Martini

    University of Birmingham, UK
  • Detlef Müller

    Christian-Albrechts-Universität zu Kiel, Germany
  • Sebastiano Nicolussi Golo

    University of Birmingham, UK; University of Jyväskylä, Finland
Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type cover
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Abstract

Let be a smooth second-order real differential operator in divergence form on a manifold of dimension . Under a bracket-generating condition, we show that the ranges of validity of spectral multiplier estimates of Mikhlin–Hörmander type and wave propagator estimates of Miyachi–Peral type for cannot be wider than the corresponding ranges for the Laplace operator on . The result applies to all sub-Laplacians on Carnot groups and more general sub-Riemannian manifolds, without restrictions on the step. The proof hinges on a Fourier integral representation for the wave propagator associated with and nondegeneracy properties of the sub-Riemannian geodesic flow.

Cite this article

Alessio Martini, Detlef Müller, Sebastiano Nicolussi Golo, Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type. J. Eur. Math. Soc. 25 (2023), no. 3, pp. 785–843

DOI 10.4171/JEMS/1191