On the constants of the Bohnenblust-Hille inequality and Hardy--Littlewood inequalities

Abstract

In this paper, among other results, we improve the best known estimates for the constants of the generalized Bohnenblust-Hille inequality. These enhancements are then used to improve the best known constants of the Hardy--Littlewood inequality; this inequality asserts that for a positive integer $m\geq2$ with $2m\leq p\leq\infty$ and $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ there exists a constant $C_{m,p}^{\mathbb{K}}\geq1$ such that, for all continuous $m$--linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$, and all positive integers $n$,% \[ \left( \sum_{j_{1},...,j_{m}=1}^{n}\left\vert T(e_{j_{1}},...,e_{j_{m}% })\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq C_{m,p}^{\mathbb{K}}\left\Vert T\right\Vert , \] and the exponent $\frac{2mp}{mp+p-2m}$ is sharp. In particular, we show that for $p > 2m^{3}-4m^{2}+2m$ the optimal constants satisfying the above inequality are dominated by the best known estimates for the constants of the $m$-linear Bohnenblust--Hille inequality. More precisely if $\gamma$ denotes the Euler--Mascheroni constant, considering the case of complex scalars as an illustration, we show that% \[ C_{m,p}^{\mathbb{C}}\leq\prod\limits_{j=2}^{m}\Gamma\left( 2-\frac{1}% {j}\right) ^{\frac{j}{2-2j}}<m^{\frac{1-\gamma}{2}}, \] which is somewhat surprising since this new formula has no dependence on $p$ (the former estimate depends on $p$ but, paradoxally, is worse than this new one). This suggests the following open problems: 1) Are the optimal constants of the Hardy--Littlewood inequality and Bohnenblust--Hille inequalities the same? 2) Are the optimal constants of the Hardy--Littlewood inequality independent of $p$ (at least for large $p$)?