Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators
Annales de l'Institut Fourier, Volume 31 (1981) no. 4, pp. 115-140.

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.

On étudie les sommes de Riesz pour les développements en fonctions propres pour une classe d’opérateurs hypoelliptiques sur le groupe de Heisenberg. Les opérateurs que l’on considère sont homogènes et invariants par l’action du gorupe unitaire. On obtient des résultats de convergence en norme L p , aux points de Lebesgue et presque partout. On prouve aussi des résultats de localisation.

@article{AIF_1981__31_4_115_0,
     author = {Mauceri, Giancarlo},
     title = {Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators},
     journal = {Annales de l'Institut Fourier},
     pages = {115--140},
     publisher = {Institut Fourier},
     volume = {31},
     number = {4},
     year = {1981},
     doi = {10.5802/aif.851},
     zbl = {0455.35039},
     mrnumber = {84h:35125},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.851/}
}
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
DA  - 1981///
SP  - 115
EP  - 140
VL  - 31
IS  - 4
PB  - Institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.851/
UR  - https://zbmath.org/?q=an%3A0455.35039
UR  - https://www.ams.org/mathscinet-getitem?mr=84h:35125
UR  - https://doi.org/10.5802/aif.851
DO  - 10.5802/aif.851
LA  - en
ID  - AIF_1981__31_4_115_0
ER  - 
%0 Journal Article
%A Mauceri, Giancarlo
%T Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators
%J Annales de l'Institut Fourier
%D 1981
%P 115-140
%V 31
%N 4
%I Institut Fourier
%U https://doi.org/10.5802/aif.851
%R 10.5802/aif.851
%G en
%F AIF_1981__31_4_115_0
Mauceri, Giancarlo. Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators. Annales de l'Institut Fourier, Volume 31 (1981) no. 4, pp. 115-140. doi : 10.5802/aif.851. http://archive.numdam.org/articles/10.5802/aif.851/

[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). | Zbl

[2] J. L. Clerc, Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, Ann. Inst. Fourier, Grenoble, 24, 1 (1974), 149-172. | Numdam | MR | Zbl

[3] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. | MR | Zbl

[4] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[7] D. Geller, Fourier analysis on the Heisenberg group I : Schwartz space, J. Funct. Anal., 36 (1980), 205-254. | MR | Zbl

[8] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. | MR | Zbl

[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] R. A. Kunze, Lp Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc., 89 (1958), 519-540. | MR | Zbl

[11] G. Mauceri, Zonal multipliers on the Heisenberg group, Pacific J. Math. (to appear). | Zbl

[12] G. Metivier, Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques, Comm. Part. Differential Equations, 1 (1976), 467-519. | MR | Zbl

[13] J. Peetre, Remark on eigenfunction expansions for elliptic operators with constant coefficients, Math. Scand., 15 (1964), 83-92. | MR | Zbl

[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. | Zbl

[15] C. Rockland, Hypoellipticity on the Heisenberg group : representation theoretic criteria, Trans. Amer. Math. Soc., 240 (1978), 1-52. | MR | Zbl

Cited by Sources: