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arXiv 1803.01397math.FA
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多重线性型上Hardy-Littlewood不等式的泛界

Universal bounds for the Hardy--Littlewood inequalities on multilinear forms

Gustavo Araújo, Kleber Câmara

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摘要：

序列空间上多重线性型的Hardy-Littlewood不等式指出，对于所有正整数$m,n\geq2$和所有$m$ -线性型$T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$（$\mathbb{K}=\mathbb{R}$或$\mathbb{C}$），存在常数$C_{m}\geq1$（不依赖于$n$）使得\[ \left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^ρ\right) ^{\frac{1}ρ}\leq C_{m}\sup_{\left\Vert x_{1}\right\Vert ,\dots,\left\Vert x_{m}\right\Vert \leq 1}\left\vert T(x_{1},\dots,x_{m})\right\vert, \]，其中如果$0\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\leq\frac{1}{2}$，则$ρ=\frac{2m}{m+1-2\left( \frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\right) }$，如果$\frac{1}{2}\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}<1$，则$ρ=\frac{1}{1-\left( \frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\right)}$。哈迪-利特尔伍德常数的良好估计通常与数学甚至物理学中的应用有关，但这些常数的确切行为仍然未知。本文对$\frac{1}{2}\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}<1$情形下常数的性质作了一些新的贡献。作为主要结果的结果，我们给出了Aron等人关于某些Hardy-Littlewood型不等式的一个结果的推广和简化证明。

英文摘要：

The Hardy--Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers $m,n\geq2$ and all $m$-linear forms $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) there are constants $C_{m}\geq1$ (not depending on $n$) such that \[ \left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^ρ\right) ^{\frac{1}ρ}\leq C_{m}\sup_{\left\Vert x_{1}\right\Vert ,\dots,\left\Vert x_{m}\right\Vert \leq 1}\left\vert T(x_{1},\dots,x_{m})\right\vert, \] where $ρ=\frac{2m}{m+1-2\left( \frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\right) }$ if $0\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\leq\frac{1}{2}$ or $ρ=\frac{1}{1-\left( \frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\right)}$ if $\frac{1}{2}\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}<1$. Good estimates for the Hardy-Littlewood constants are, in general, associated to applications in Mathematics and even in Physics, but the exact behavior of these constants is still unknown. In this note we give some new contributions to the behavior of the constants in the case $\frac{1}{2}\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}<1$. As a consequence of our main result, we present a generalization and a simplified proof of a result due to Aron et al. on certain Hardy--Littlewood type inequalities.

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