Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators
Annales de l'Institut Fourier, Volume 31 (1981) no. 4, p. 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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {31},
     number = {4},
     year = {1981},
     pages = {115-140},
     doi = {10.5802/aif.851},
     zbl = {0455.35039},
     mrnumber = {84h:35125},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_1981__31_4_115_0/

[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 0093.06901

[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 50 #14065 | Zbl 0273.22011

[3] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. | MR 56 #6264 | Zbl 0358.30023

[4] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. | MR 58 #13215 | Zbl 0312.35026

[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 51 #3719 | Zbl 0293.35012

[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 17,158d | Zbl 0058.08802

[7] D. Geller, Fourier analysis on the Heisenberg group I : Schwartz space, J. Funct. Anal., 36 (1980), 205-254. | MR 81g:43008 | Zbl 0433.43008

[8] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. | MR 26 #2986 | Zbl 0111.18101

[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 20 #6668 | Zbl 0084.33905

[11] G. Mauceri, Zonal multipliers on the Heisenberg group, Pacific J. Math. (to appear). | Zbl 0474.43009

[12] G. Metivier, Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques, Comm. Part. Differential Equations, 1 (1976), 467-519. | MR 55 #888 | Zbl 0376.35012

[13] J. Peetre, Remark on eigenfunction expansions for elliptic operators with constant coefficients, Math. Scand., 15 (1964), 83-92. | MR 31 #2510 | Zbl 0131.09802

[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 0148.13002

[15] C. Rockland, Hypoellipticity on the Heisenberg group : representation theoretic criteria, Trans. Amer. Math. Soc., 240 (1978), 1-52. | MR 58 #6071 | Zbl 0326.22007