Spectral multipliers and wave equation for sub-Laplacians : lower regularity bounds of Euclidean type
Abstract
Let L be a smooth second-order real differential operator in divergence form on a manifold of dimension n. 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 L cannot be wider than the corresponding ranges for the Laplace operator on Rn. 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 L and nondegeneracy properties of the sub-Riemannian geodesic flow.
Main Authors
Format
Articles
Research article
Published
2023
Series
Subjects
Publication in research information system
Publisher
European Mathematical Society - EMS - Publishing House GmbH
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202307064423Use this for linking
Review status
Peer reviewed
ISSN
1435-9855
DOI
https://doi.org/10.4171/JEMS/1191
Language
English
Published in
Journal of the European Mathematical Society
Citation
- Martini, A., Müller, D., & Nicolussi Golo, S. (2023). Spectral multipliers and wave equation for sub-Laplacians : lower regularity bounds of Euclidean type. Journal of the European Mathematical Society, 25(3), 785-843. https://doi.org/10.4171/JEMS/1191
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Additional information about funding
This research was partially supported by the EPSRC Grant “Sub-Elliptic Harmonic Analysis”(EP/P002447/1). Part of the work was carried out during a two-month visit of the first-named author to the Christian-Albrechts-Universität zu Kiel (Germany), made possible by the generous financial support of the Alexander von Humboldt Foundation
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