Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866909218691350528 |
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| author | Martini, Alessio Müller, Detlef Golo, Sebastiano Nicolussi |
| author_facet | Martini, Alessio Müller, Detlef Golo, Sebastiano Nicolussi |
| contents | Let $\mathscr{L}$ 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 Mihlin--Hörmander type and wave propagator estimates of Miyachi--Peral type for $\mathscr{L}$ cannot be wider than the corresponding ranges for the Laplace operator on $\mathbb{R}^n$. 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 $\mathscr{L}$ and nondegeneracy properties of the sub-Riemannian geodesic flow. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1812_02671 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type Martini, Alessio Müller, Detlef Golo, Sebastiano Nicolussi Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35L05, 35S30, 42B15, 43A22, 58J60 Let $\mathscr{L}$ 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 Mihlin--Hörmander type and wave propagator estimates of Miyachi--Peral type for $\mathscr{L}$ cannot be wider than the corresponding ranges for the Laplace operator on $\mathbb{R}^n$. 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 $\mathscr{L}$ and nondegeneracy properties of the sub-Riemannian geodesic flow. |
| title | Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35L05, 35S30, 42B15, 43A22, 58J60 |
| url | https://arxiv.org/abs/1812.02671 |