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

The Hardy–Littlewood inequality for $m$-linear forms on ${\mathrm{\ell }}_p$ 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}\rightarrow $\mathbb{R}$$or$$\mathbb{C}$.$Thecase$m=2$ 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 $2^{\frac{\left( m-1\right) \left( p-m\right) }{p}}$\mathrm{.}Inparticular,for$m<p\leq m+1$ the optimal constants of the above inequality are uniformly bounded by $2.$