The optimal constants for the real Hardy--Littlewood inequality for bilinear forms on $c_{0}\times\ell_{p}$

TL;DR

This paper determines the exact constants in the Hardy--Littlewood inequality for real bilinear forms on certain sequence spaces, providing new sharp bounds and extending classical results like Littlewood's theorem.

Contribution

It precisely computes the optimal constants for the Hardy--Littlewood inequality on $c_0 imes ext{l}_p$ spaces, including new cases and improved bounds.

Findings

Exact constants $C_{p, ext{infty}}=2^{rac{1}{2}-rac{1}{p}}$ for $p o ext{large}$.

Almost optimal bounds for $2<p<2.18$, with precision better than $4 imes10^{-4}$.

Extension of Littlewood's $4/3$ theorem to new parameter ranges.

Abstract

For $p,q\geq2$, the Hardy and Littlewood inequalities for real bilinear forms, in its unified formulation, assert that there is a constant $C_{p,q}\geq1$ 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) ^{\frac{\lambda}{2}}\right) ^{\frac {1}{\lambda}}\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 $A:\ell_{p}\times\ell_{q}\rightarrow\mathbb{R}$ (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 $p\geq\frac{p_{0}}{p_{0}-1}\approx2.18.$ The number $p_{0}\in(1,2)$ is the unique real number satisfying \[ \Gamma\left(\frac{p_{0}+1}{2}\right) =\frac{\sqrt{\pi}}{2}. \] 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 $4\cdot10^{-4}$. 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.