A short communication on the constants of the multilinear Hardy--Littlewood inequality

TL;DR

This paper derives new upper bounds for the constants in the multilinear Hardy--Littlewood inequality on ll_{p} spaces for specific p ranges, improving previous estimates for all m  3.

Contribution

It provides improved upper bounds for the Hardy--Littlewood inequality constants when 2m  p  2m^{3}-4m^{2}+2m, extending and refining earlier results.

Findings

New upper bounds for the constants in the Hardy--Littlewood inequality.

Improved estimates over previous results from 2014 for all m  3.

Applicable for p in the range 2m  p  2m^{3}-4m^{2}+2m.

Abstract

It was recently proved that for $p>2m^{3}-4m^{2}+2m$ the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$-spaces are less than or equal to the best known estimates of respective constants of the Bohnenblust--Hille inequality. In this note we obtain upper bounds for opposite side, i.e., the constants when $2m\leq p\leq2m^{3}-4m^{2}+2m.$ For these values of $p$ our result improves previous estimates from 2014 of Araujo \textit{et al}. for all $m\geq3.$