A.e. convergence of spectral sums on Lie groups

[Convergence p.p. de sommes spectrales sur les groupes de Lie]

Meaney, Christopher ; Müller, Detlef ; Prestini, Elena

Annales de l'Institut Fourier, Tome 57 (2007), p. 1509-1520 / Harvested from Numdam

Résumé

Soit $\mathcal{L}$ un sous-Laplacien invariant à droite sur un groupe de Lie $G,$ et soit $S_Rf:= \int _0^R dE_\lambda f,\ R\ge 0,$ l’opérateur “sommes sphériques partielles” associé, où $\mathcal{L}=\int _0^\infty \lambda \, dE_\lambda $ dénote la résolution spectrale de $\mathcal{L}.$ Nous prouvons que $S_Rf(x)$ converge vers $f(x)$ p.p. quand $R\rightarrow \infty ,$ si $\log (2+\mathcal{L})f\in L^2(G).$

Abstract

Let $\mathcal{L}$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let $S_Rf:= \int _0^R dE_\lambda f,\ R\ge 0,$ denote the associated “spherical partial sums,” where $\mathcal{L}=\int _0^\infty \lambda \, dE_\lambda $ is the spectral resolution of $\mathcal{L}.$ We prove that $S_Rf(x)$ converges a.e. to $f(x)$ as $R\rightarrow \infty $ under the assumption $\log (2+\mathcal{L})f\in L^2(G).$

Détail

Publié le : 2007-01-01

MR 2364139 | Zbl 1131.22007

DOI : https://doi.org/10.5802/aif.2303
Classification:  22E30,  43A50
Mots clés: théorème de Rademacher-Menchov, sous-Laplacien, théorie spectrale

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