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 $\mathrm{supp}\mu \subset Q$, $\mu (Q)=0$, $\mathrm{\mid }\mu \mathrm{\mid }(Q)\le 1$, and, denoting by $p_t$ the heat kernel in ${\mathbb{R}}^d$, $$\underset{x\in {\mathbb{R}}^d,t>0}{\sup }\mathrm{\mid }{t}^{1\mathrm{/}2}{p}_t\ast \mu (x)\mathrm{\mid }\le \frac{1}{l(Q{)}^{d-1}}\mathrm{.}$$ 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.