Summability of multilinear forms on classical sequence spaces

TL;DR

This paper extends the Hardy--Littlewood inequality to multilinear forms on classical sequence spaces, providing optimal bounds and exponents for different parameter ranges, thus broadening the understanding of summability in multilinear analysis.

Contribution

The paper introduces new bounds and optimal exponents for the summability of multilinear forms on sequence spaces, extending previous inequalities to more general settings.

Findings

Established bounds for multilinear forms on ll_p spaces.

Proved the optimality of the exponents in the inequalities.

Unified previous results as special cases of the new inequalities.

Abstract

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.