A closed formula for subexponential constants in the multilinear Bohnenblust--Hille inequality

Abstract

For the scalar field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, the multilinear Bohnenblust--Hille inequality asserts that there exists a sequence of positive scalars $(C_{\mathbb{K},m})_{m=1}^{\infty}$ such that %[(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}}%,...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{\mathbb{K},m}\sup_{z_{1},...,z_{m}\in\mathbb{D}^{N}}|U(z_{1},...,z_{m})|] for all $m$-linear form $U:\mathbb{K}^{N}\times...\times\mathbb{K}% ^{N}\rightarrow\mathbb{K}$ and every positive integer $N$, where $(e_{i})_{i=1}^{N}$ denotes the canonical basis of $\mathbb{K}^{N}$ and $\mathbb{D}^{N}$ represents the open unit polydisk in $\mathbb{K}^{N}$. Since its proof in 1931, the estimates for $C_{\mathbb{K},m}$ have been improved in various papers. In 2012 it was shown that there exist constants $(C_{\mathbb{K},m})_{m=1}^{\infty}$ with subexponential growth satisfying the Bohnenblust-Hille inequality. However, these constants were obtained via a complicated recursive formula. In this paper, among other results, we obtain a closed (non-recursive) formula for these constants with subexponential growth.