# A.e. convergence of spectral sums on Lie groups

• [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)
• Volume: 57, Issue: 5, page 1509-1520
• ISSN: 0373-0956

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## Abstract

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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 $log\left(2+ℒ\right)f\in {L}^{2}\left(G\right).$

## How to cite

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Meaney, Christopher, Müller, Detlef, and Prestini, Elena. "A.e. convergence of spectral sums on Lie groups." Annales de l’institut Fourier 57.5 (2007): 1509-1520. <http://eudml.org/doc/10267>.

@article{Meaney2007,
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 $\log (2+\mathcal\{L\})f\in L^2(G).$},
affiliation = {Macquarie University Department of Mathematics North Ryde NSW 2109 (Australia); C.A.-Universität Kiel Mathematisches Seminar Ludewig-Meyn-Str.4 D-24098 Kiel (Germany); Università di Roma “Tor Vergata" Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italie)},
author = {Meaney, Christopher, Müller, Detlef, Prestini, Elena},
journal = {Annales de l’institut Fourier},
keywords = {Rademacher-Menshov theorem; sub-Laplacian; spectral theory; sub-Laplacian spectral theory},
language = {eng},
number = {5},
pages = {1509-1520},
publisher = {Association des Annales de l’institut Fourier},
title = {A.e. convergence of spectral sums on Lie groups},
url = {http://eudml.org/doc/10267},
volume = {57},
year = {2007},
}

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
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 5
SP - 1509
EP - 1520
AB - 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 $\log (2+\mathcal{L})f\in L^2(G).$
LA - eng
KW - Rademacher-Menshov theorem; sub-Laplacian; spectral theory; sub-Laplacian spectral theory
UR - http://eudml.org/doc/10267
ER -

## References

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6. A. Hulanicki, J. W. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), 703-715 Zbl0516.43010MR701519
7. J. Ludwig, D. Müller, Sub-Laplacians of holomorphic ${L}^{p}$-type on rank one $AN$-groups and related solvable groups, J. Funct. Anal. 170 (2000), 366-427 Zbl0957.22013MR1740657
8. N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, 100 (1992), Cambridge University Press, Cambridge Zbl0813.22003MR1218884
9. A. Zygmund, Trigonometric Series, 1 and 2 (2002), Cambridge Mathematical Library. Cambridge University Press, Cambridge Zbl1084.42003MR1963498

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