The optimal constants for the real Hardy--Littlewood inequality for bilinear forms on $c_0\times {\ell }_p$

Abstract

For $p,q\ge 2$, the Hardy and Littlewood inequalities for real bilinear forms, in its unified formulation, assert that there is a constant $C_{p,q}\ge 1$ such that \begin{equation} \left(\sum\limits_{j=1}^{\infty}\left(\sum\limits_{k=1}^{\infty}\left\vert A(e_{j},e_{k})\right\vert ^{2}\right) ^{\fracλ{2}}\right) ^{\frac {1}λ}\leq C_{p,q}\left\Vert A\right\Vert, \end{equation} with sharp exponent $\lambda =\frac{pq}{pq-p-q},$ for all continuous bilinear forms $A:{\ell }_p\times {\ell }_q\to \mathbb{R}$ (as usual, $c_0$ replaces ${\ell }_p$ or ${\ell }_q$ when $p=\infty$ or $q=\infty$)$.$ In this note, among other results, we show that the sharp constants $C_{p,\infty }$ are precisely \[ C_{p,\infty}=2^{\frac{1}{2}-\frac{1}{p}}% \] whenever $p\ge \frac{p_0}{p_0-1}\approx \mathrm{2.18.}$ The number $p_0\in (1,2)$ is the unique real number satisfying \[ Γ\left(\frac{p_{0}+1}{2}\right) =\frac{\sqrtπ}{2}. \] In the remaining case, i.e., for $2<p<\frac{p_0}{p_0-1}\approx 2.18,$ we obtain almost optimal constants, with better precision than $4\cdot 10^{-4}$. This last result extends a result from Diniz et al. giving the sharp constant of the famous Littlewood's $4/3$ theorem for real scalars.

This paper has been withdrawn. The reason is that it is now superceded by the paper arXiv:1604.06323(v2)

References in corpus (2)

• On the upper bounds for the constants of the Hardy-Littlewood inequality
• A note on the Bohnenblust--Hille inequality for multilinear forms