Sommes de Riesz et multiplicateurs sur un groupe de Lie compact
Clerc, Jean-Louis
Annales de l'Institut Fourier, Tome 24 (1974), p. 149-172 / Harvested from Numdam
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On étudie diverses convergences des sommes de Riesz des fonctions de puissance pième sommable sur un groupe de Lie compact. On montre que n-1 2, où n est la dimension du groupe, est un indice critique pour la classe L 1 . On donne également un théorème de multiplicateurs qui redonne le résultat classique de Marcinkiewicz pour le tore. On établit enfin un lien entre les multiplicateurs des groupes de Lie compacts et certains multiplicateurs de R n .

Convergence of Riesz means of p-summable functions are studied extensively. Explicitly n-1 2 is shown to be a critical index for L 1 convergence. We prove a multiplier theorem which reduces to Marcinkiewicz’s result on the 1-torus. We also find a link between compact Lie groups multipliers and some multipliers of R n (n= the dimension of the group).

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     author = {Clerc, Jean-Louis},
     title = {Sommes de Riesz et multiplicateurs sur un groupe de Lie compact},
     journal = {Annales de l'Institut Fourier},
     volume = {24},
     year = {1974},
     pages = {149-172},
     doi = {10.5802/aif.496},
     mrnumber = {50 \#14065},
     zbl = {0273.22011},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1974__24_1_149_0}
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Clerc, Jean-Louis. Sommes de Riesz et multiplicateurs sur un groupe de Lie compact. Annales de l'Institut Fourier, Tome 24 (1974) pp. 149-172. doi : 10.5802/aif.496. http://gdmltest.u-ga.fr/item/AIF_1974__24_1_149_0/

[1] S. Bochner, Summation of multiple Fourier series by spherical means. Trans. Amer. Math. Soc., 40 (1936), 175-207. | JFM 62.0293.03 | MR 1501870 | Zbl 0015.15702

[2] A. Bonami et J.L. Clerc, Sommes de Cesaro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc., 183 (1973), 223-263. | MR 49 #3461 | Zbl 0278.43015

[3] Chandrasekharan et Minakshisundaram, Typical means, Bombay 1952. | MR 14,1077c | Zbl 0047.29901

[4] R. Coifman et G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in mathematics, n° 242. Springer Verlag (1971). | MR 58 #17690 | Zbl 0224.43006

[5a] C. Fefferman, Inequalities for strongly singular convolution operators. Acta Math., 124 (1970), 9-36. | MR 41 #2468 | Zbl 0188.42601

[5b] C. Fefferman, The multiplier problem for the ball, Amer. J. Math., 94 (1971), 330-336. | MR 45 #5661 | Zbl 0234.42009

[6] S. Helgason, Differential geometry and symmetric spaces, Acad. Press New-York (1962). | MR 26 #2986 | Zbl 0111.18101

[7a] L. Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, Recent Advances in the Basic Sciences, Yeshiva University Conference, Nov. 1966, 155-202.

[7b] L. Hörmander. The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218. | MR 58 #29418 | Zbl 0164.13201

[8] Kung Sun, Fourier analysis on unitary groups, V ; spherical summability and Fourier integrals, Chinese Math., 7 (1965), 1-20.

[9] Séminaire S. Lie.

[10a] E.M. Sein, Localization and summability of multiple Fourier series, Acta Math., 100 (1958), 93-147. | MR 21 #4331 | Zbl 0085.28401

[10b] E.M. Stein, Topics in harmonic analysis. Annals of Math. Studies 63. Princeton Univ. Press, New-Jersey (1970). | Zbl 0193.10502

[10c] E.M. Stein, On certain exponential sums arising in multiple Fourier series, Ann. of Math., 73 (1961), 87-109. | MR 23 #A2715 | Zbl 0099.05501

[11] G. Szegö, Orthogonal polynomials, Amer Math. Soc. Coll. Publ. n° 23, 1939. | JFM 65.0278.03 | Zbl 0023.21505

[12] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer-Verlag 1972. | Zbl 0265.22020

[13] N. Weiss, Lp-estimates for bi-invariant operators on compact Lie groups, Amer. J. of Math., 94 (1972), 103-118. | MR 45 #5278 | Zbl 0239.43004