From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces

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 $\mathrm{B}\mathrm{M}\mathrm{O}(Q_0)$ when $n=1$. For $n\ge 2$, 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 $\mathrm{\parallel }{\mathbf{1}}_E{\mathrm{\parallel }}_{{\mathcal{B}}_q(Q_0)}\sim \mathrm{\parallel }{\mathbf{1}}_E{\mathrm{\parallel }}_{\mathcal{B}(Q_0)}\sim \min \{1,P(E,Q_0)\}$ 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 $I_{\epsilon ,q}^{\ast }(E)$, defined using arbitrarily oriented cubes, converges to $\frac{1}{2}\min \{1,P(E,Q_0)\}$ as $\epsilon \to 0^{+}$ throughout $q\in (0,\frac{n}{n-1}]$. In dimension two, the formula extends to $q\in (0,q_{\ast }]$ for some universal $q_{\ast }>\frac{19}{9}$, whereas it fails in every dimension for every $q>q_0$, where $q_0>2$ is defined variationally and numerical computation suggests that $q_0\approx 2.15391$. 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.