A.e. convergence of spectral sums on Lie groups
Annales de l'Institut Fourier, Volume 57 (2007) no. 5, p. 1509-1520

Let $ℒ$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let ${S}_{R}f:={\int }_{0}^{R}d{E}_{\lambda }f,\phantom{\rule{4pt}{0ex}}R\ge 0,$ denote the associated “spherical partial sums,” where $ℒ={\int }_{0}^{\infty }\lambda \phantom{\rule{0.166667em}{0ex}}d{E}_{\lambda }$ is the spectral resolution of $ℒ.$ We prove that ${S}_{R}f\left(x\right)$ converges a.e. to $f\left(x\right)$ as $R\to \infty$ under the assumption $log\left(2+ℒ\right)f\in {L}^{2}\left(G\right).$

Soit $ℒ$ un sous-Laplacien invariant à droite sur un groupe de Lie $G,$ et soit ${S}_{R}f:={\int }_{0}^{R}d{E}_{\lambda }f,\phantom{\rule{4pt}{0ex}}R\ge 0,$ l’opérateur “sommes sphériques partielles” associé, où $ℒ={\int }_{0}^{\infty }\lambda \phantom{\rule{0.166667em}{0ex}}d{E}_{\lambda }$ dénote la résolution spectrale de $ℒ.$ Nous prouvons que ${S}_{R}f\left(x\right)$ converge vers $f\left(x\right)$ p.p. quand $R\to \infty ,$ si $log\left(2+ℒ\right)f\in {L}^{2}\left(G\right).$

DOI : https://doi.org/10.5802/aif.2303
Classification:  22E30,  43A50
Keywords: Rademacher-Menshov theorem, sub-Laplacian, spectral theory
@article{AIF_2007__57_5_1509_0,
author = {Meaney, Christopher and M\"uller, Detlef and Prestini, Elena},
title = {A.e. convergence of spectral sums on Lie groups},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {57},
number = {5},
year = {2007},
pages = {1509-1520},
doi = {10.5802/aif.2303},
mrnumber = {2364139},
zbl = {1131.22007},
language = {en},
url = {http://www.numdam.org/item/AIF_2007__57_5_1509_0}
}

Meaney, Christopher; Müller, Detlef; Prestini, Elena. A.e. convergence of spectral sums on Lie groups. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1509-1520. doi : 10.5802/aif.2303. http://www.numdam.org/item/AIF_2007__57_5_1509_0/

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