Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant

Abstract

The Hardy--Littlewood inequality for $m$-linear forms on ${\ell }_p$ spaces and $m<p\le 2m$ asserts that \begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*} for all continuous $m$-linear forms $T:{\ell }_p\times \cdots \times {\ell }_p\to \mathbb{R}$ or $\mathbb{C}.$ The case $m=2$ recovers a classical inequality proved by Hardy and Littlewood in 1934. As a consequence of the results of the present paper we show that the same inequality is valid with $2^{\frac{m-1}{2}}$ replaced by $2^{\frac{\left(m-1\right)\left(p-m\right)}{p}}$. In particular, for $m<p\le m+1$ the optimal constants of the above inequality are uniformly bounded by $2.$

Cited by in corpus (2)

• A new estimate for the constants of an inequality due to Hardy and Littlewood
• Some applications of the Regularity Principle in sequence spaces