Optimal Hardy-Littlewood type inequalities for $m$-linear forms on ${\mathrm{\ell }}_p$ spaces with $1\le p\le m$

The Hardy-Littlewood inequalities for $m$-linear forms on ${\mathrm{\ell }}_p$ spaces are stated for $p>m$. In this paper, among other results, we investigate similar results for $1\le p\le m\mathrm{.}$ Let $\mathbb{K}$ be $$ or $\mathbb{C}$ and $m\ge 2$ be a positive integer. Our main results are the following sharp inequalities: (i) If $\left(r,p\right) \in \left(\lbrack 1,2]\times \lbrack 2,2m)\right) \cup \left(\lbrack 1,\infty)\times \lbrack 2m,\infty \right)) $, then there is a constant $D_{m,r,p}^{\mathbb{K}}>0$ (not depending on $$) such that \begin{equation*} \textstyle\left(\sum\limits_{j_{1},…,j_{m}=1}^{n}\left\vert T(e_{j_{1}},…,e_{j_{m}})\right\vert ^{r}\right) ^{\frac{1}{r}}\leq D_{m,r,p}^{$\mathbb{K}$}n^{\max \left{ \frac{2mr+2mp-mpr-pr}{2pr},0\right} }\left\Vert T\right\Vert \end{equation*} for all $m$–linear forms $T:{\mathrm{\ell }}_p^n\times \cdots \times {\mathrm{\ell }}_p^n\to \mathbb{K}$ and all positive integers $n$. (ii) If $(r,p)\in [2,\mathrm{\infty })\times (m,2m]$, then \begin{equation*} \textstyle\left(\sum\limits_{j_{1},…,j_{m}=1}^{n}\left\vert T(e_{j_{1}},…,e_{j_{m}})\right\vert ^{r}\right) ^{\frac{1}{r}}\leq \left(\sqrt{2}\right) ^{m-1}n^{\max \left{ \frac{p+mr-rp}{pr},0\right} }\left\Vert T\right\Vert \end{equation*} for all $m$–linear forms $T:{\mathrm{\ell }}_p^n\times \cdots \times {\mathrm{\ell }}_p^n\to \mathbb{K}$ and all positive integers $n\mathrm{.}$ Moreover the exponents $\max { (2mr+2mp-mpr-pr)/2pr,0 } $in(i)and$\max {(p+mr-rp)/pr,0 } $ in (ii) are optimal.