Diana Marcela Serrano-Rodriguez

arXiv2026

The Geometry of Real Anisotropic Bohnenblust--Hille Constants

We determine the growth scale of the optimal constants in the real anisotropic Bohnenblust--Hille inequality. For an exponent vector $\mathbf q^{(m)}$, write $C_{\mathbf q^{(m)}}^{(m)}$ for its optimal constant and $d_m$ for its diameter. These constants are superpolynomial precisely when $m d_m/\log m\to\infty$; throughout this regime, their logarithm has the sharp scale $m d_m$. If also $d_m\to0$, then $\log C_{\mathbf q^{(m)}}^{(m)}/(m d_m)$ lies asymptotically in the interval $\left[\frac{\log 2}{4},\frac{2-\log 2-γ}{4}\right]$, where $γ$ is the Euler--Mascheroni constant; this interval has width less than $10^{-2}$. We also solve the extremal problems at fixed diameter and fixed total deficit. The normalized reciprocal-deficit profiles order the canonically arranged optimal constants by majorization, yield exact formulas on a full-dimensional region, and recover the lower coefficient $(\log 2)/4$ throughout a broad class of anisotropic regimes.

math.FA↗

arXiv2012

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

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.

math.FA↗