The optimal constants of the mixed $\left({\ell }_1,{\ell }_2\right)$-Littlewood inequality

Abstract

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 $\left(1,2,...,2\right)$, sometimes called mixed $\left({\ell }_1,{\ell }_2\right)$-Littlewood inequality. We show that these optimal constants are precisely ${\left(\sqrt{2}\right)}^{m-1}$ 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 Albuquerque \textit{et al.} in a paper published in 2014 in the \textit{Journal of Functional Analysis}.

accepted for publication in the Journal of Number Theory

References in corpus (2)

• On the upper bounds for the constants of the Hardy-Littlewood inequality
• On the constants of the Bohnenblust-Hille inequality and Hardy--Littlewood inequalities