The optimal constants for the real Hardy--Littlewood inequality for bilinear forms on c₀timesell_(p)

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.