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

Summability and estimates for polynomials and multilinear mappings

AbstractIn this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ℓp spaces in fact hold true for mappings on arbitrary Banach spaces