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

Abstract

The Hardy-Littlewood inequalities for $m$-linear forms on ${\ell }_p$ spaces are stated for $p>m$. In this paper, among other results, we investigate similar results for $1\le p\le m.$ 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 $\left(r,p\right)\in \left([1,2]\times [2,2m)\right)\cup \left([1,\infty )\times [2m,\infty \right))$, 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 $T:{\ell }_p^n\times \cdots \times {\ell }_p^n\to \mathbb{K}$ and all positive integers $n$. (ii) If $\left(r,p\right)\in [2,\infty )\times (m,2m]$, 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 $T:{\ell }_p^n\times \cdots \times {\ell }_p^n\to \mathbb{K}$ and all positive integers $n.$ Moreover the exponents $\max \{(2mr+2mp-mpr-pr)/2pr,0\}$ in (i) and $\max \{(p+mr-rp)/pr,0\}$ in (ii) are optimal.

References in corpus (3)

• On the upper bounds for the constants of the Hardy-Littlewood inequality
• On the constants of the Bohnenblust-Hille inequality and Hardy--Littlewood inequalities
• Optimal estimates for summing multilinear operators

Cited by in corpus (1)

• A note on multiple summing operators and applications