On the Constants of the Bohnenblust–Hille and Hardy–Littlewood Inequalities

Abstract

For $\mathbb {K}=\mathbb {R}$ or $\mathbb {C}$, the Hardy–Littlewood inequality for m-linear forms asserts that for $4\le 2m\le p\le \infty$ there exists a constant $C_{m,p}^{\mathbb {K}}\ge 1$ such that, for all m-linear forms $T:\ell _{p}^{n}\times \cdots \times \ell _{p}^{n}\rightarrow \mathbb {K}$, and all positive integers n,

This result was proved by Hardy and Littlewood (QJ Math 5:241–254, 1934) for bilinear forms and extended to m-linear forms by Praciano-Pereira (J Math Anal Appl 81:561–568, 1981). The case $p=\infty$ recovers the Bohnenblust–Hille inequality (Ann Math 32:600–622, 1931). In this paper, among other results, we show that for $p>2m(m-1)^2$ the optimal constants satisfying the Hardy–Littlewood inequality for m-linear forms are dominated by the best known constants of the corresponding Bohnenblust–Hille inequality. For instance, we show that if $p>2m(m-1)^2$, then

$$\begin{aligned} \textstyle C_{m,p}^{\mathbb {C}}\le \prod \limits _{j=2}^{m}\Gamma \left( 2-\frac{1}{j}\right) ^{\frac{j}{2-2j}}<m^{\frac{1-\gamma }{2}}, \end{aligned}$$

where $\gamma$ is the Euler–Mascheroni constant.