A Cohen type inequality for compact Lie groups

Giulini, Saverio; Soardi, Paolo M.; Travaglini, Giancarlo

doi:10.1090/S0002-9939-1979-0545596-3

A Cohen type inequality for compact Lie groups
HTML articles powered by AMS MathViewer

by Saverio Giulini, Paolo M. Soardi and Giancarlo Travaglini PDF

Proc. Amer. Math. Soc. 77 (1979), 359-364 Request permission

Abstract:

The following theorem is proved: let G denote a compact connected semisimple Lie group. There exists $\theta = \theta (G)(3 \leqslant \theta < 4)$ such that, if ${\chi _1}, \ldots ,{\chi _N}$ are N distinct characters of G, ${d_1}, \ldots ,{d_N}$ their dimensions, ${c_1}, \ldots ,{c_N}$ complex numbers of modulus greater than or equal to one, then, for all $p > \theta ,|||\Sigma _{j = 1}^N{c_j}{d_j}{\chi _j}||{|_p} \geqslant \text {const}_p N^{\alpha _p}$ where $||| \cdot ||{|_p}$ denotes the ${L^p}(G)$ convolutor norm and $\text {const}_p$ and ${\alpha _p} = {\alpha _p}(G)$ are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.

References

J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0252560
Robert S. Cahn, Lattice points and Lie groups. I, II, Trans. Amer. Math. Soc. 183 (1973), 119–129; ibid. 183 (1973), 131–137. MR 335687, DOI 10.1090/S0002-9947-1973-0335687-3
Carlo Cecchini, Lacunary Fourier series on compact Lie groups, J. Functional Analysis 11 (1972), 191–203. MR 0374821, DOI 10.1016/0022-1236(72)90088-2
Jean-Louis Clerc, Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 1, 149–172 (French). MR 361620
Paul J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191–212. MR 133397, DOI 10.2307/2372731
A. H. Dooley, Norms of characters and lacunarity for compact Lie groups, J. Functional Analysis 32 (1979), no. 2, 254–267. MR 534677, DOI 10.1016/0022-1236(79)90057-0
Edwin Hewitt and Herbert S. Zuckerman, On a theorem of P. J. Cohen and H. Davenport, Proc. Amer. Math. Soc. 14 (1963), 847–855. MR 154947, DOI 10.1090/S0002-9939-1963-0154947-1
S. K. Pichorides, A lower bound for the $L^{1}$ norm of exponential sums, Mathematika 21 (1974), 155–159. MR 371831, DOI 10.1112/S0025579300008536

On a conjecture of Littlewood concerning exponential sums

24

J. F. Price, On the integral divergence of Dirichlet kernels for the second unitary and special unitary groups, J. London Math. Soc. (2) 9 (1974/75), 593–598. MR 390661, DOI 10.1112/jlms/s2-9.4.593
J. F. Price, On local central lacunary sets for compact Lie groups, Monatsh. Math. 80 (1975), no. 3, 201–204. MR 390659, DOI 10.1007/BF01319915
Robert J. Stanton, Mean convergence of Fourier series on compact Lie groups, Trans. Amer. Math. Soc. 218 (1976), 61–87. MR 420158, DOI 10.1090/S0002-9947-1976-0420158-9
Robert J. Stanton and Peter A. Tomas, Polyhedral summability of Fourier series on compact Lie groups, Amer. J. Math. 100 (1978), no. 3, 477–493. MR 622197, DOI 10.2307/2373834
E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. MR 125392, DOI 10.2307/1970308
V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0376938