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The optimal constants of the mixed \((\ell_{1},\ell_{2})\)-Littlewood inequality. (English) Zbl 1431.46024
Summary: In this note, among other results, we find the optimal constants of the generalized Bohnenblust-Hille inequality for \(m\)-linear forms over \(\mathbb{R}\) and with multiple exponents \((1, 2, \dots, 2)\), sometimes called mixed \((\ell_1, \ell_2)\)-Littlewood inequality. We show that these optimal constants are precisely \((\sqrt{2})^{m-1}\) and this is somewhat surprising since a series of recent papers have shown that similar constants have a sublinear growth. This result answers a question raised by N. Albuquerque et al. in [J. Funct. Anal. 266, No. 6, 3726–3740 (2014; Zbl 1319.46035)].

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
26D15 Inequalities for sums, series and integrals
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