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RESEARCH PRODUCT
Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
Alessio MartiniMüller Detlef Sebastiano Nicolussi Golosubject
osittaisdifferentiaaliyhtälötsub-LaplacianApplied MathematicsGeneral Mathematicsharmoninen analyysi35L05 35S30 42B15 43A22 58J60Functional Analysis (math.FA)Mathematics - Functional Analysiseikonal equationMathematics - Analysis of PDEsMathematics - Classical Analysis and ODEsClassical Analysis and ODEs (math.CA)FOS: Mathematicswave equationsub-Riemannian manifoldMathematics::Differential Geometryspectral multipliermonistotFourier integral operatorAnalysis of PDEs (math.AP)description
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--H\"ormander 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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-12-06 | Journal of the European Mathematical Society |