A.e. convergence of spectral sums on Lie groups

Meaney, Christopher; Müller, Detlef; Prestini, Elena

doi:10.5802/aif.2303

A.e. convergence of spectral sums on Lie groups
[Convergence p.p. de sommes spectrales sur les groupes de Lie]

Meaney, Christopher ¹ ; Müller, Detlef ² ; Prestini, Elena ³

¹ Macquarie University Department of Mathematics North Ryde NSW 2109 (Australia)
² C.A.-Universität Kiel Mathematisches Seminar Ludewig-Meyn-Str.4 D-24098 Kiel (Germany)
³ Università di Roma “Tor Vergata" Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italie)

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1509-1520.

Résumé
Abstract

Soit $ℒ$ un sous-Laplacien invariant à droite sur un groupe de Lie $G,$ et soit $S_{R} f : = \int_{0}^{R} d E_{λ} f, R \geq 0,$ l’opérateur “sommes sphériques partielles” associé, où $ℒ = \int_{0}^{\infty} λ d E_{λ}$ dénote la résolution spectrale de $ℒ .$ Nous prouvons que $S_{R} f (x)$ converge vers $f (x)$ p.p. quand $R \to \infty,$ si $log (2 + ℒ) f \in L^{2} (G) .$

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

MR Zbl

DOI : 10.5802/aif.2303

Classification : 22E30, 43A50
Keywords: Rademacher-Menshov theorem, sub-Laplacian, spectral theory
Mot clés : théorème de Rademacher-Menchov, sous-Laplacien, théorie spectrale

Affiliations des auteurs :

Meaney, Christopher ¹ ; Müller, Detlef ² ; Prestini, Elena ³

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     title = {A.e. convergence of spectral sums on {Lie} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1509--1520},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
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Meaney, Christopher; Müller, Detlef; Prestini, Elena. A.e. convergence of spectral sums on Lie groups. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1509-1520. doi : 10.5802/aif.2303. http://www.numdam.org/articles/10.5802/aif.2303/

Bibliographie
Cité par

[1] Alexits, G. Convergence Problems of Orthogonal Series., International Series of Monographs in Pure and Applied Mathematics, 20, Pergamon Press, Oxford, New York, 1961 (Translated from the German by I. Földer) | MR | Zbl

[2] Carbery, A.; Soria, F. Almost-Everywhere Convergence of Fourier Integrals for Functions in Sobolev Spaces, and an $L^{2}$ -Localisation Principle, Rev. Mat. Iberoamericana, Volume 4 (1988) no. 2, pp. 319-337 | MR | Zbl

[3] Christ, M. $L^{p}$ bounds for spectral multipliers on nilpotent groups., Trans. Amer. Math. Soc., Volume 328 (1991) no. 1, pp. 73-81 | DOI | MR | Zbl

[4] Colzani, L.; Meaney, C.; Prestini, E. Almost everywhere convergence of inverse Fourier transforms., Proc. Amer. Math. Soc., Volume 134 (2006) no. 6, pp. 1651-1660 | DOI | MR | Zbl

[5] Folland, G. B.; Stein, E. M. Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28, Princeton University Press, Princeton, N.J., 1982 | MR | Zbl

[6] Hulanicki, A.; Jenkins, J. W. Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc., Volume 278 (1983) no. 2, pp. 703-715 | DOI | MR | Zbl

[7] Ludwig, J.; Müller, D. Sub-Laplacians of holomorphic $L^{p}$ -type on rank one $A N$ -groups and related solvable groups, J. Funct. Anal., Volume 170 (2000) no. 2, pp. 366-427 | DOI | MR | Zbl

[8] Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, T. Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[9] Zygmund, A. Trigonometric Series, 1 and 2, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002 (With a foreword by Robert A. Fefferman) | MR | Zbl

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