Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
Alessio MartiniUniversity of Birmingham, UK
Detlef MüllerChristian-Albrechts-Universität zu Kiel, Germany
Sebastiano Nicolussi GoloUniversity of Birmingham, UK; University of Jyväskylä, Finland
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. 785–843DOI 10.4171/JEMS/1191