An atomic decomposition for functions of bounded variation

In this paper, we give a decomposition of the gradient measure [Formula: see text] of an arbitrary function of bounded variation [Formula: see text] into a linear combination of atoms [Formula: see text], where [Formula: see text] is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each [Formula: see text], there exists a cube [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], and, denoting by [Formula: see text] the heat kernel in [Formula: see text], [Formula: see text] 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.