On the upper bounds for the constants of the Hardy-Littlewood inequality

TL;DR

This paper improves upper bounds for the constants in the Hardy-Littlewood inequality for m-linear forms on b5_p spaces, showing subpolynomial growth when p 65; m^2, which advances understanding of these bounds.

Contribution

The authors provide new estimates for the Hardy-Littlewood inequality constants that depend on p and m, improving upon the previous dcb1d estimates.

Findings

New bounds depend on p and m, improving previous estimates.

Constants grow subpolynomially when p 65; m^2.

Results refine understanding of Hardy-Littlewood inequality constants.

Abstract

The best known upper estimates for the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$ spaces are of the form $\left(\sqrt{2}\right) ^{m-1}.$ We present better estimates which depend on $p$ and $m$. An interesting consequence is that if $p\geq m^{2}$ then the constants have a subpolynomial growth as $m$ tends to infinity.