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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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References:
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