从Bourgain--Brezis--Mironescu空间中的尖锐自改进到精确周长公式
From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces
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中文总结 AI 辅助
本文研究Bourgain--Brezis--Mironescu空间的$L^q$变体，刻画了函数与特征函数的端点行为，并建立了精确周长公式的收敛性及其维度限制。
中文摘要 AI 辅助
设$Q_0:=(0,1)^n$。我们通过将Bourgain--Brezis--Mironescu空间$\mathcal{B}(Q_0)$中的局部$L^1$平均振荡替换为其$L^q$对应物来定义$\mathcal{B}_q(Q_0)$，当$n=1$时该空间与$\mathrm{BMO}(Q_0)$一致。对于$n\geq2$，我们的结果揭示了一般函数、特征函数和精确周长公式的不同端点行为。(i) 对于一般函数，$\mathcal{B}_q(Q_0)=\mathcal{B}(Q_0)$当且仅当$q\in(0,\frac{n}{n-1})$。我们还为Hardy型前对偶建立了相应的尖锐自改进和对偶结果。(ii) 对于特征函数，端点达到：等价关系$\\|\mathbf{1}_E\\|_{\mathcal{B}_q(Q_0)}\sim\\|\mathbf{1}_E\\|_{\mathcal{B}(Q_0)}\sim\min\{1,P(E,Q_0)\}$对可测集$E\subset Q_0$一致成立当且仅当$q\in(0,\frac{n}{n-1}]$，其中$P(E,Q_0)$表示相对周长。(iii) 各向同性泛函$I_{\varepsilon,q}^{*}(E)$（使用任意定向立方体定义）在$q\in(0,\frac{n}{n-1}]$范围内当$\varepsilon\to0^+$时收敛到$\frac{1}{2}\min\{1,P(E,Q_0)\}$。在二维情形，该公式扩展到$q\in(0,q_*]$，其中$q_*>\frac{19}{9}$为某个普适常数；而在每个维度中，对于所有$q>q_0$该公式失效，其中$q_0>2$通过变分方式定义，数值计算表明$q_0\approx2.15391$。三个尖锐自改进的证明分别使用稀疏中位数支配、原子块分解和相对等周不等式。对于周长公式，一个一致的De Giorgi爆破将问题归结为几何立方截面不等式；几何平均和显式填充分别给出平面扩展和障碍。
英文摘要
Let $Q_0:=(0,1)^n$. We define $\mathcal{B}_q(Q_0)$ by replacing the local $L^1$ mean oscillation in the Bourgain--Brezis--Mironescu space $\mathcal{B}(Q_0)$ with its $L^q$ counterpart, which coincides with $\mathrm{BMO}(Q_0)$ when $n=1$. For $n\geq2$, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, $\mathcal{B}_q(Q_0)=\mathcal{B}(Q_0)$ precisely when $q\in(0,\frac{n}{n-1})$. We also establish the corresponding sharp self-improvement and duality results for Hardy-type preduals. (ii) For characteristic functions, the endpoint is attained: the equivalence $\|\mathbf{1}_E\|_{\mathcal{B}_q(Q_0)}\sim\|\mathbf{1}_E\|_{\mathcal{B}(Q_0)}\sim\min\{1,P(E,Q_0)\}$ holds uniformly over measurable $E\subset Q_0$ precisely when $q\in(0,\frac{n}{n-1}]$, where $P(E,Q_0)$ denotes relative perimeter. (iii) The isotropic functional $I_{\varepsilon,q}^{*}(E)$, defined using arbitrarily oriented cubes, converges to $\frac{1}{2}\min\{1,P(E,Q_0)\}$ as $\varepsilon\to0^+$ throughout $q\in(0,\frac{n}{n-1}]$. In dimension two, the formula extends to $q\in(0,q_*]$ for some universal $q_*>\frac{19}{9}$, whereas it fails in every dimension for every $q>q_0$, where $q_0>2$ is defined variationally and numerical computation suggests that $q_0\approx2.15391$. The proofs of the three sharp self-improvements use sparse median domination, atomic block decompositions, and relative isoperimetry, respectively. For the perimeter formula, a uniform De Giorgi blow-up reduces the problem to a geometric cube-section inequality; geometric averaging and an explicit packing yield the planar extension and the obstruction, respectively.
