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An atomic decomposition for functions of bounded variation

Daniel Spector, Cody B. Stockdale, Dmitriy Stolyarov

TL;DR

This work develops an atomic framework for BV functions by decomposing the gradient measure Du into sums of (d−1)-atoms supported on dyadic cubes, built from Dχ_F with controlled size and heat-kernel normalization. A key Sampling Lemma expresses ∇u for smooth u as limits of sums of derivatives of characteristic functions, while a Boxing inequality enables a dyadic boxing of sets of finite perimeter. The main result extends to the BV setting on ℝ^d via density arguments, yielding a weak-* representation of [Du]_l as sums of atoms with uniformly controlled total variation, and leading to Sobolev-type inequalities, dimension estimates, and trace inequalities; these are further linked to DS_beta Besov–Lorentz-type spaces and provide a unifying view of BV analysis with operator-cancellation insights. The approach clarifies how atomic decompositions interact with dimension and trace phenomena and offers a pathway to generalize these estimates to other differential operators.

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 $μ=Dχ_{F}$, 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 $\operatorname*{supp}μ\subset Q$, $μ(Q)=0$, $|μ|(Q)\leq 1$, and, denoting by $p_t$ the heat kernel in $\mathbb{R}^d$, \[ \sup_{x \in \mathbb{R}^d, t>0} |t^{1/2} p_t \ast μ(x)| \leq \frac{1}{l(Q)^{d-1}}. \] 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.

An atomic decomposition for functions of bounded variation

TL;DR

This work develops an atomic framework for BV functions by decomposing the gradient measure Du into sums of (d−1)-atoms supported on dyadic cubes, built from Dχ_F with controlled size and heat-kernel normalization. A key Sampling Lemma expresses ∇u for smooth u as limits of sums of derivatives of characteristic functions, while a Boxing inequality enables a dyadic boxing of sets of finite perimeter. The main result extends to the BV setting on ℝ^d via density arguments, yielding a weak-* representation of [Du]_l as sums of atoms with uniformly controlled total variation, and leading to Sobolev-type inequalities, dimension estimates, and trace inequalities; these are further linked to DS_beta Besov–Lorentz-type spaces and provide a unifying view of BV analysis with operator-cancellation insights. The approach clarifies how atomic decompositions interact with dimension and trace phenomena and offers a pathway to generalize these estimates to other differential operators.

Abstract

In this paper, we give a decomposition of the gradient measure of an arbitrary function of bounded variation into a sum of atoms , where is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each , there exists a cube such that , , , and, denoting by the heat kernel in , 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.
Paper Structure (3 sections, 9 theorems, 61 equations)

This paper contains 3 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. A Sampling Lemma
  3. Proofs of Main Results

Key Result

Theorem 1.2

Let $u \in \dot{\mathop{\mathrm{BV}}\nolimits}(\mathbb{R}^d)$. There exist sets of finite perimeter $\{E_{i,n}\}_{n \in \mathbb{N}, i=1,\ldots,n}$, dyadic cubes $\{Q_{i,n}\}_{n \in \mathbb{N}, i=1,\ldots,n}$ with $Q_{i_1,n} \cap Q_{i_2,n} = \emptyset$ for each $n \in \mathbb{N}$ and $i_1\neq i_2$, a is a $(d-1)$-atom, in the weak--star topology of the space of measures, and

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Lemma 1.14
  • ...and 5 more