实各向异性Bohnenblust--Hille常数的几何

中文摘要 AI 辅助

我们确定了实各向异性Bohnenblust--Hille不等式最优常数的增长尺度。对于指数向量$\boldsymbol{q}^{(m)}$，记其最优常数为$C_{\boldsymbol{q}^{(m)}}^{(m)}$，直径为$d_m$。当$m d_m/\boldsymbol{\text{log}} \boldsymbol{m}\to\boldsymbol{\text{∞}}$时，这些常数呈超多项式增长；在此范围内，其对数的精确尺度为$m d_m$。若同时$d_m\to0$，则$\boldsymbol{\text{log}} C_{\boldsymbol{q}^{(m)}}^{(m)}/(m d_m)$渐近位于区间$\boldsymbol{\text{[}}\frac{\boldsymbol{\text{log}} 2}{4}\boldsymbol{,}\frac{\boldsymbol{2 - \text{log}} \boldsymbol{2 - \boldsymbol{\text{γ}}}}{4}\boldsymbol{]}$，其中$\boldsymbol{\text{γ}}$为欧拉-马歇罗尼常数，该区间宽度小于$10^{-2}$。我们还求解了固定直径和固定总亏格下的极值问题，归一化倒数亏格轮廓通过优化排序规范排列的最优常数，在全维区域给出精确公式，并在广泛的各向异性范围内恢复下限系数$\frac{\boldsymbol{\text{log}} 2}{4}$。

英文摘要

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.

发表机构

• Universidad Nacional de Colombia（哥伦比亚国立大学）
• Universidade Federal da Paraíba（帕拉伊巴联邦大学）
• Oklahoma State University（俄克拉荷马州立大学）

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