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 linear combination of atoms μ = DχF, where F is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each μ, there exists a cube Q such that supp μ ∪ Q, μ(Q) = 0, |μ|(Q) ≤ 1, and, denoting by pt the heat kernel in ℝ^d, esssup_xϵℝ^d,t>0|t^1/2pt ∗ μ(x)|≤ 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.