A.e. convergence of spectral sums on Lie groups
[Convergence p.p. de sommes spectrales sur les groupes de Lie]
Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1509-1520.

Soit un sous-Laplacien invariant à droite sur un groupe de Lie G, et soit S R f:= 0 R dE λ f,R0, l’opérateur “sommes sphériques partielles” associé, où = 0 λdE λ dénote la résolution spectrale de . Nous prouvons que S R f(x) converge vers f(x) p.p. quand R, si log(2+)fL 2 (G).

Let be a right-invariant sub-Laplacian on a connected Lie group G, and let S R f:= 0 R dE λ f,R0, denote the associated “spherical partial sums,” where = 0 λdE λ is the spectral resolution of . We prove that S R f(x) converges a.e. to f(x) as R under the assumption log(2+)fL 2 (G).

DOI : https://doi.org/10.5802/aif.2303
Classification : 22E30,  43A50
Mots clés : théorème de Rademacher-Menchov, sous-Laplacien, théorie spectrale
@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},
     pages = {1509--1520},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     doi = {10.5802/aif.2303},
     mrnumber = {2364139},
     zbl = {1131.22007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2303/}
}
TY  - JOUR
AU  - Meaney, Christopher
AU  - Müller, Detlef
AU  - Prestini, Elena
TI  - A.e. convergence of spectral sums on Lie groups
JO  - Annales de l'Institut Fourier
PY  - 2007
DA  - 2007///
SP  - 1509
EP  - 1520
VL  - 57
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2303/
UR  - https://www.ams.org/mathscinet-getitem?mr=2364139
UR  - https://zbmath.org/?q=an%3A1131.22007
UR  - https://doi.org/10.5802/aif.2303
DO  - 10.5802/aif.2303
LA  - en
ID  - AIF_2007__57_5_1509_0
ER  - 
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://archive.numdam.org/articles/10.5802/aif.2303/

[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 218827 | Zbl 0098.27403

[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 1028744 | Zbl 0692.42001

[3] Christ, M. L p bounds for spectral multipliers on nilpotent groups., Trans. Amer. Math. Soc., Volume 328 (1991) no. 1, pp. 73-81 | Article | MR 1104196 | Zbl 0739.42010

[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 | Article | MR 2204276 | Zbl 1082.42006

[5] Folland, G. B.; Stein, E. M. Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28, Princeton University Press, Princeton, N.J., 1982 | MR 657581 | Zbl 0508.42025

[6] Hulanicki, A.; Jenkins, J. W. Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc., Volume 278 (1983) no. 2, pp. 703-715 | Article | MR 701519 | Zbl 0516.43010

[7] Ludwig, J.; Müller, D. Sub-Laplacians of holomorphic L p -type on rank one AN-groups and related solvable groups, J. Funct. Anal., Volume 170 (2000) no. 2, pp. 366-427 | Article | MR 1740657 | Zbl 0957.22013

[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 1218884 | Zbl 0813.22003

[9] Zygmund, A. Trigonometric Series, 1 and 2, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002 (With a foreword by Robert A. Fefferman) | MR 1963498 | Zbl 1084.42003

Cité par Sources :