An atomic decomposition for functions of bounded variation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910927577677824 |
|---|---|
| author | Spector, Daniel Stockdale, Cody B. Stolyarov, Dmitriy |
| author_facet | Spector, Daniel Stockdale, Cody B. Stolyarov, Dmitriy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02053 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An atomic decomposition for functions of bounded variation Spector, Daniel Stockdale, Cody B. Stolyarov, Dmitriy Functional Analysis Analysis of PDEs 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. |
| title | An atomic decomposition for functions of bounded variation |
| topic | Functional Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2505.02053 |