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
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by Saverio Giulini, Paolo M. Soardi and Giancarlo Travaglini

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

DOI: https://doi.org/10.1090/S0002-9939-1979-0545596-3

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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.

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