A.e. convergence of spectral sums on Lie groups

Christopher Meaney[1]; Detlef Müller[2]; Elena Prestini[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 (2007)

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

Abstract

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Let ℒ be a right-invariant sub-Laplacian on a connected Lie group G , and let S R f : = ∫ 0 R d E λ f , R ≥ 0 , denote the associated “spherical partial sums,” where ℒ = ∫ 0 ∞ 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 → ∞ under the assumption log ( 2 + ℒ ) f ∈ L 2 ( G ) .

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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  3. M. Christ, L p bounds for spectral multipliers on nilpotent groups., Trans. Amer. Math. Soc. 328 (1991), 73-81 Zbl0739.42010MR1104196
  4. L. Colzani, C. Meaney, E. Prestini, Almost everywhere convergence of inverse Fourier transforms., Proc. Amer. Math. Soc. 134 (2006), 1651-1660 Zbl1082.42006MR2204276
  5. G. B. Folland, E. M. Stein, Hardy Spaces on Homogeneous Groups, 28 (1982), Princeton University Press, Princeton, N.J. Zbl0508.42025MR657581
  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 A N -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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