The optimal constants of the mixed $\left( \ell_{1},\ell _{2}\right) $-Littlewood inequality

TL;DR

This paper determines the exact optimal constants for the mixed ,-Littlewood inequality in the context of m-linear forms over the real numbers, revealing they grow exponentially as ^{m-1}, contrary to previous beliefs of sublinear growth.

Contribution

It provides the first explicit calculation of the optimal constants for the generalized Bohnenblust--Hille inequality with mixed exponents, resolving a question from 2014.

Findings

Optimal constants are ()^{m-1} for the inequality.

Contradicts previous assumptions of sublinear growth of similar constants.

Answers a longstanding open question in the field.

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}.