Critical Hardy--Littlewood inequality for multilinear forms

The Hardy–Littlewood inequalities for $m$-linear forms on ${\mathrm{\ell }}_p$ spaces are known just for $p>m$. The critical case $p=m$ was overlooked for obvious technical reasons and, up to now, the only known estimate is the trivial one. In this paper we deal with this critical case of the Hardy–Littlewood inequality. More precisely, for all positive integers $m\ge 2$ we have $$\underset{j_1}{\sup }{\left(\sum _{j_2=1}^n{\left(\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}{\left(\sum _{j_m=1}^n{\mid T(e_{j_1},\dots ,e_{j_m})\mid }^{s_m}\right)}^{\frac{1}{s_m}\cdot s_{m-1}}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\right)}^{\frac{1}{s_3}{s}_2}\right)}^{\frac{1}{s_2}}\le 2^{\frac{m-2}{2}}\parallel T\parallel$$ for all $m$–linear forms $T:\ell_{m}^{n}\times\cdots\times\ell_{m} ^{n}\rightarrow$\mathbb{K}$=$\mathbb{R}$$or$$\mathbb{C}$$with$s_{k} =\frac{2m(m-1)}{m+mk-2k}$forall$k=2,…,m$andforallpositiveintegers$n$. As a corollary, for the classical case of bilinear forms investigated by Hardy and Littlewood in 1934 our result is sharp in a strong sense (both exponents and constants are optimal for real and complex scalars).