An atomic decomposition for functions of bounded variation

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 $\hat{I}¼=D\ddot{I}{‡}_F$, where $F$ is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each $\hat{I}¼$, there exists a cube $Q$ such that $\mathrm{supp}\hat{I}¼\subset Q$, $\hat{I}¼(Q)=0$, $|\hat{I}¼|(Q)\le 1$, and, denoting by $p_t$ the heat kernel in ${\mathbb{R}}^d$, \[ \sup_{x \in \mathbb{R}^d, t>0} |t^{1/2} p_t \ast μ(x)| \leq \frac{1}{l(Q)^{d-1}}. \] 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.

13 pages