Sommes de Riesz et multiplicateurs sur un groupe de Lie compact

Clerc, Jean-Louis

Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 149-172.

On étudie diverses convergences des sommes de Riesz des fonctions de puissance pième sommable sur un groupe de Lie compact. On montre que $\frac{n-1}{2}$, 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 $\frac{n-1}{2}$ 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).

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

[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 | Zbl

[3] Chandrasekharan et Minakshisundaram, Typical means, Bombay 1952. | MR | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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