Summability of multilinear forms on classical sequence spaces

We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\geq k\,$ and $n_{1},\dots ,n_{k}$ be positive integers such that $n_{1}+\cdots +n_{k}=m$. ($a$) If $(r,p)\in (0,\infty )\times \lbrack 2m,\infty ]$ then there is a constant $D_{m,r,p,k}^{\mathbb{K}}\geq 1$ (not depending on $n$) such that \left( \sum_{i_{1},\dots ,i_{k}=1}^{n}\left| T\left( e_{i_{1}}^{n_{1}},\dots ,e_{i_{k}}^{n_{k}}\right) \right| ^{r}\right) ^{% \frac{1}{r}}\leq D_{m,r,p,k}^{\mathbb{K}} \cdot n^{max\left\{ \frac{% 2kp-kpr-pr+2rm}{2pr},0\right\} }\left| T\right| for all $m$-linear forms $T:\ell_{p}^{n}\times \cdots \times \ell_{p}^{n}\rightarrow \mathbb{K}$ and all positive integers $n$. Moreover, the exponent $max\left\{ \frac{2kp-kpr-pr+2rm}{2pr},0\right\}$ is optimal. ($b$) If $(r, p) \in (0, \infty) \times (m, 2m]$ then there is a constant $D_{m,r,p, k}^{\mathbb{K}}\geq 1$ (not depending on $n$) such that \left( \sum_{i_{1},\dots ,i_{k}=1}^{n }\left| T\left( e_{i_{1}}^{n_{1}},\dots ,e_{i_{k}}^{n_{k}}\right) \right| ^{r }\right) ^{% \frac{1}{r }}\leq D_{m,r,p, k}^{\mathbb{K}} \cdot n^{ max \left\{\frac{% p-rp+rm}{pr}, 0\right\}}\left| T\right| for all $m$-linear forms $T:\ell_{p}^{n}\times \cdots \times \ell_{p}^{n}\rightarrow \mathbb{K}$ and all positive integers $n$. Moreover, the exponent $max \left\{\frac{p-rp+rm}{pr}, 0\right\}$ is optimal. The case $k=m$ recovers a recent result due to G. Araujo and D. Pellegrino