An atomic decomposition for functions of bounded variation

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.