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

Authors: Saverio Giulini, Paolo M. Soardi and Giancarlo Travaglini
Journal: Proc. Amer. Math. Soc. 77 (1979), 359-364
MSC: Primary 43A55; Secondary 22E30, 43A50
DOI: https://doi.org/10.1090/S0002-9939-1979-0545596-3
MathSciNet review: 545596
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $p > \theta ,\vert\vert\vert\Sigma _{j = 1}^N{c_j}{d_j}{\chi _j}\vert\vert{\vert _p} \geqslant$ const $_p N^{\alpha_p}$ where $\vert\vert\vert \cdot \vert\vert{\vert _p}$ denotes the ${L^p}(G)$ convolutor norm and const and ${\alpha _p} = {\alpha _p}(G)$ are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.

[1] J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0252560
[2] Robert S. Cahn, Lattice points and Lie groups. I, II, Trans. Amer. Math. Soc. 183 (1973), 119–129; ibid. 183 (1973), 131–137. MR 0335687, https://doi.org/10.1090/S0002-9947-1973-0335687-3
[3] Carlo Cecchini, Lacunary Fourier series on compact Lie groups, J. Functional Analysis 11 (1972), 191–203. MR 0374821
[4] 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 0361620
[5] Paul J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191–212. MR 0133397, https://doi.org/10.2307/2372731
[6] A. H. Dooley, Norms of characters and lacunarity for compact Lie groups, J. Funct. Anal. 32 (1979), no. 2, 254–267. MR 534677, https://doi.org/10.1016/0022-1236(79)90057-0
[7] 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 0154947, https://doi.org/10.1090/S0002-9939-1963-0154947-1
[8] S. K. Pichorides, A lower bound for the 𝐿¹ norm of exponential sums, Mathematika 21 (1974), 155–159. MR 0371831, https://doi.org/10.1112/S0025579300008536
[9] -, On a conjecture of Littlewood concerning exponential sums, Notices Amer. Math. Soc. 24 (1977), A-427.
[10] 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 0390661, https://doi.org/10.1112/jlms/s2-9.4.593
[11] J. F. Price, On local central lacunary sets for compact Lie groups, Monatsh. Math. 80 (1975), no. 3, 201–204. MR 0390659, https://doi.org/10.1007/BF01319915
[12] Robert J. Stanton, Mean convergence of Fourier series on compact Lie groups, Trans. Amer. Math. Soc. 218 (1976), 61–87. MR 0420158, https://doi.org/10.1090/S0002-9947-1976-0420158-9
[13] 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 0622197, https://doi.org/10.2307/2373834
[14] E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. MR 0125392, https://doi.org/10.2307/1970308
[15] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Prentice-Hall Series in Modern Analysis. MR 0376938