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arXiv 1609.03081math.FA
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一致有界的最优Hardy-Littlewood不等式

Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant

N. Albuquerque, G. Araújo, M. Maia, T. Nogueira, D. Pellegrino, J. Santos

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摘要：

$m$ -线性形式在$\ell _{p}$空间和$m<p\leq 2m$上的Hardy-Littlewood不等式断言：对于所有连续的$m$ -线性形式，\begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*}为$T:\ell _{p}\times \cdots \times \ell _{p}\rightarrow \mathbb{R}$或$\mathbb{C}.$。这种情形$m=2$恢复了Hardy和Littlewood在1934年证明的一个经典不等式。作为本文结果的结果，我们表明，同样的不等式是有效的$2^{\frac{m-1}{2}}$取代$2^{\frac{\left( m-1\right) \left( p-m\right) }{p}}$。特别地，对于$m<p\leq m+1$，上述不等式的最优常数由$2.$一致有界

英文摘要：

The Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces and $m<p\leq 2m$ asserts that \begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*} for all continuous $m$-linear forms $T:\ell _{p}\times \cdots \times \ell _{p}\rightarrow \mathbb{R}$ or $\mathbb{C}.$ The case $m=2$ recovers a classical inequality proved by Hardy and Littlewood in 1934. As a consequence of the results of the present paper we show that the same inequality is valid with $2^{\frac{m-1}{2}}$ replaced by $2^{\frac{\left( m-1\right) \left( p-m\right) }{p}}$. In particular, for $m<p\leq m+1$ the optimal constants of the above inequality are uniformly bounded by $2.$

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