A Counterexample to a Bourgain-Brezis Sum-Space Estimate
Abstract
A question of Bourgain and Brezis, later restated as Open Problem 7.1 by Brezis, asks whether the H1/2+W1,1 norm of a mean-zero scalar function is controlled by the ambient H−1/2+L1 sum norm of its gradient. We show that this estimate fails in every dimension N≥ 2. More precisely, for every integer K≥ 16, we construct a smooth real-valued mean-zero function ΦK on 𝕋N such that |∇ΦK|H−1/2(𝕋N)+L1(𝕋N)≤ C and |ΦK|H1/2(𝕋N)+W1,1(𝕋N)≥ cK1/4−C, where the absolute constants c,C>0 are independent of N and K. Equivalently, the ratio of the gradient-constrained decomposition cost to the ambient decomposition cost is unbounded. The construction applies the Hodge projection to a smooth approximation of a normal vector measure on a codimension-one sheet. Its potential records lacunary partial sums at geometrically separated distances from the sheet. A scale-path lower bound, combined with an odd critical Hardy inequality, excludes uniformly bounded H1/2+W1,1 decompositions.
Related
- The Bourgain, Brezis, Mironescu localization for deterministic obstacle problems
- Homogenization and averaging in the Bourgain-Brezis-Mironescu limit
- Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
- Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
- From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces
BibTeX
@misc{zhang2026counterexample,
title = {{A Counterexample to a Bourgain-Brezis Sum-Space Estimate}},
author = {Chenye Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35882},
eprint = {2610.35882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35882},
}
RIS
TY - UNPB
TI - A Counterexample to a Bourgain-Brezis Sum-Space Estimate
AU - Chenye Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.35882
UR - https://arxiv.org/abs/2610.35882
AB - A question of Bourgain and Brezis, later restated as Open Problem 7.1 by Brezis, asks whether the H^1/2+W^1,1 norm of a mean-zero scalar function is controlled by the ambient H^−1/2+L^1 sum norm of its gradient. We show that this estimate fails in every dimension N≥ 2. More precisely, for every integer K≥ 16, we construct a smooth real-valued mean-zero function Φ_K on 𝕋^N such that |∇Φ_K|H^−1/2(𝕋^N)+L^1(𝕋^N)≤ C and |Φ_K|H^1/2(𝕋^N)+W^1,1(𝕋^N)≥ cK^1/4−C, where the absolute constants c,C>0 are independent of N and K. Equivalently, the ratio of the gradient-constrained decomposition cost to the ambient decomposition cost is unbounded. The construction applies the Hodge projection to a smooth approximation of a normal vector measure on a codimension-one sheet. Its potential records lacunary partial sums at geometrically separated distances from the sheet. A scale-path lower bound, combined with an odd critical Hardy inequality, excludes uniformly bounded H^1/2+W^1,1 decompositions.
ER -