From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces
Abstract
Let Q0:=(0,1)n. We define ℬq(Q0) by replacing the local L1 mean oscillation in the Bourgain--Brezis--Mironescu space ℬ(Q0) with its Lq counterpart, which coincides with BMO(Q0) when n=1. For n≥2, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, ℬq(Q0)=ℬ(Q0) precisely when q∈(0,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 |1E|ℬq(Q0)∼|1E|ℬ(Q0)∼min{1,P(E,Q0)} holds uniformly over measurable E⊂ Q0 precisely when q∈(0,n/(n−1)], where P(E,Q0) denotes relative perimeter. (iii) The isotropic functional Iε,q*(E), defined using arbitrarily oriented cubes, converges to 1/2min{1,P(E,Q0)} as ε→0+ throughout q∈(0,n/(n−1)]. In dimension two, the formula extends to q∈(0,q*] for some universal q*>19/9, whereas it fails in every dimension for every q>q0, where q0>2 is defined variationally and numerical computation suggests that q0≈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.
Related
- The Bourgain, Brezis, Mironescu localization for deterministic obstacle problems
- Homogenization and averaging in the Bourgain-Brezis-Mironescu limit
- Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
- Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
- A Counterexample to a Bourgain-Brezis Sum-Space Estimate
BibTeX
@misc{jia2026from,
title = {{From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces}},
author = {Hongchao Jia and Xianjie Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04849},
eprint = {2610.04849},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04849},
}
RIS
TY - UNPB
TI - From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces
AU - Hongchao Jia
AU - Xianjie Yan
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04849
UR - https://arxiv.org/abs/2610.04849
AB - Let Q_0:=(0,1)^n. We define ℬ_q(Q_0) by replacing the local L^1 mean oscillation in the Bourgain--Brezis--Mironescu space ℬ(Q_0) with its L^q counterpart, which coincides with BMO(Q_0) when n=1. For n≥2, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, ℬ_q(Q_0)=ℬ(Q_0) precisely when q∈(0,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 |1_E|_ℬ_q(Q_0)∼|1_E|_ℬ(Q_0)∼min{1,P(E,Q_0)} holds uniformly over measurable E⊂ Q_0 precisely when q∈(0,n/(n−1)], where P(E,Q_0) denotes relative perimeter. (iii) The isotropic functional I_ε,q^*(E), defined using arbitrarily oriented cubes, converges to 1/2min{1,P(E,Q_0)} as ε→0^+ throughout q∈(0,n/(n−1)]. In dimension two, the formula extends to q∈(0,q_*] for some universal q_*>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≈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.
ER -