Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators

Giancarlo Mauceri

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 4, page 115-140
  • ISSN: 0373-0956

Abstract

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We study the Riesz means for the eigenfunction expansions of a class of hypoelliptic differential operators on the Heisenberg group. The operators we consider are homogeneous with respect to dilations and invariant under the action of the unitary group. We obtain convergence results in L p norm, at Lebesgue points and almost everywhere. We also prove localization results.

How to cite

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Mauceri, Giancarlo. "Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators." Annales de l'institut Fourier 31.4 (1981): 115-140. <http://eudml.org/doc/74511>.

@article{Mauceri1981,
abstract = {We study the Riesz means for the eigenfunction expansions of a class of hypoelliptic differential operators on the Heisenberg group. The operators we consider are homogeneous with respect to dilations and invariant under the action of the unitary group. We obtain convergence results in $L^p$ norm, at Lebesgue points and almost everywhere. We also prove localization results.},
author = {Mauceri, Giancarlo},
journal = {Annales de l'institut Fourier},
keywords = {Riesz means; eigenfunction expansions; hypoelliptic differential operators; Heisenberg group; dilations; Lebesgue points; localization results},
language = {eng},
number = {4},
pages = {115-140},
publisher = {Association des Annales de l'Institut Fourier},
title = {Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators},
url = {http://eudml.org/doc/74511},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Mauceri, Giancarlo
TI - Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 4
SP - 115
EP - 140
AB - We study the Riesz means for the eigenfunction expansions of a class of hypoelliptic differential operators on the Heisenberg group. The operators we consider are homogeneous with respect to dilations and invariant under the action of the unitary group. We obtain convergence results in $L^p$ norm, at Lebesgue points and almost everywhere. We also prove localization results.
LA - eng
KW - Riesz means; eigenfunction expansions; hypoelliptic differential operators; Heisenberg group; dilations; Lebesgue points; localization results
UR - http://eudml.org/doc/74511
ER -

References

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  1. [1] G. BERGENDAL, Convergence and summability of eigenfunction expansions connected with elliptic differential operators, Thesis, Lund, 1959, (Medd. Lunds Univ. Mat. Sem. 14, 1-63 (1959). Zbl0093.06901
  2. [2] J. L. CLERC, Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, Ann. Inst. Fourier, Grenoble, 24, 1 (1974), 149-172. Zbl0273.22011MR50 #14065
  3. [3] R. COIFMAN and G. WEISS, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. Zbl0358.30023MR56 #6264
  4. [4] G. B. FOLLAND, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. Zbl0312.35026MR58 #13215
  5. [5] G. B. FOLLAND and E. M. STEIN, Estimates for the zb complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522. Zbl0293.35012MR51 #3719
  6. [6] L. GÅRDING, On the asymptotic properties of the spectral function belonging to a self-adjoint semi-bounded extension of an elliptic differential operator, Kungl. Fysiogr. Sällsk. i Lund Förh., 24, 21 (1954), 1-18. Zbl0058.08802MR17,158d
  7. [7] D. GELLER, Fourier analysis on the Heisenberg group I : Schwartz space, J. Funct. Anal., 36 (1980), 205-254. Zbl0433.43008MR81g:43008
  8. [8] S. HELGASON, Differential geometry and symmetric spaces, Academic Press, New York, 1962. Zbl0111.18101MR26 #2986
  9. [9] L. HÖRMANDER, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, Recent Advances in Basic Sciences, Yeshiva University Conference, Nov. 1966, 155-202. 
  10. [10] R. A. KUNZE, Lp Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc., 89 (1958), 519-540. Zbl0084.33905MR20 #6668
  11. [11] G. MAUCERI, Zonal multipliers on the Heisenberg group, Pacific J. Math. (to appear). Zbl0474.43009
  12. [12] G. METIVIER, Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques, Comm. Part. Differential Equations, 1 (1976), 467-519. Zbl0376.35012MR55 #888
  13. [13] J. PEETRE, Remark on eigenfunction expansions for elliptic operators with constant coefficients, Math. Scand., 15 (1964), 83-92. Zbl0131.09802MR31 #2510
  14. [14] J. PEETRE, Some remarks on continuous orthogonal expansions, and eigenfunction expansions for positive self-adjoint elliptic operators with variable coefficients, Math. Scand., 17 (1965), 56-64. Zbl0148.13002
  15. [15] C. ROCKLAND, Hypoellipticity on the Heisenberg group : representation theoretic criteria, Trans. Amer. Math. Soc., 240 (1978), 1-52. Zbl0326.22007MR58 #6071

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