Year: 2026
Type: preprint
Abstract: 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 $N\geq 2$. More precisely, for every integer $K\geq 16$, we construct a smooth real-valued mean-zero function $Φ_K$ on $\mathbb{T}^N$ such that $|\nablaΦ_K|{H^{-1/2}(\mathbb{T}^N)+L^1(\mathbb{T}^N)}\leq C$ and $|Φ_K|{H^{1/2}(\mathbb{T}^N)+W^{1,1}(\mathbb{T}^N)}\geq cK^{1/4}-C$, 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.
Source: arXiv (Cornell University)
Author: Zhang, Chenye
Institution: Peking University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224107381