An atomic decomposition for functions of bounded variation

TL;DR

This paper introduces a novel atomic decomposition of the gradient measure of functions of bounded variation, enabling new insights into Sobolev inequalities, dimension estimates, and trace inequalities.

Contribution

It provides a new atomic decomposition framework for BV functions' gradients, utilizing a sampling of the coarea formula and a boxing identity.

Findings

Decomposition of $Du$ into atoms with specific support and size conditions

Establishment of Sobolev inequalities for BV functions

Derivation of dimension and trace inequalities

Abstract

In this paper, we give a decomposition of the gradient measure $Du$ of an arbitrary function of bounded variation $u$ into a sum of atoms $\mu=D\chi_{F}$, where $F$ is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each $\mu$, there exists a cube $Q$ such that $\operatorname*{supp}\mu\subset Q$, $\mu(Q)=0$, $|\mu|(Q)\leq 1$, and, denoting by $p_t$ the heat kernel in $\mathbb{R}^d$, \[ \sup_{x \in \mathbb{R}^d, t>0} |t^{1/2} p_t \ast \mu (x)| \leq \frac{1}{l(Q)^{d-1}}. \] Our proof relies on a sampling of the coarea formula and a new boxing identity. We present several consequences of this result, including Sobolev inequalities, dimension estimates, and trace inequalities.