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Weighted Bounded Variation Revisited

Simon Bortz, Matthew Gossett, Joseph Kasel, Kabe Moen

Abstract

In this article, we investigate the theory of weighted functions of bounded variation (BV), as introduced by Baldi [Ba01]. Depending on the theorem, we impose lower semicontinuity and/or a pointwise A1 condition on the weight. Our motivation is twofold: to establish weighted Gagliardo-Nirenberg-Sobolev (GNS) inequalities for BV functions, and to clarify and extend earlier results on weighted BV spaces. Our main contributions include a structure theorem under minimal assumptions (lower semicontinuity), a smooth approximation result, an embedding theorem, a weighted GNS inequality for BV functions, and a corresponding weighted isoperimetric inequality.

Weighted Bounded Variation Revisited

Abstract

In this article, we investigate the theory of weighted functions of bounded variation (BV), as introduced by Baldi [Ba01]. Depending on the theorem, we impose lower semicontinuity and/or a pointwise A1 condition on the weight. Our motivation is twofold: to establish weighted Gagliardo-Nirenberg-Sobolev (GNS) inequalities for BV functions, and to clarify and extend earlier results on weighted BV spaces. Our main contributions include a structure theorem under minimal assumptions (lower semicontinuity), a smooth approximation result, an embedding theorem, a weighted GNS inequality for BV functions, and a corresponding weighted isoperimetric inequality.
Paper Structure (14 sections, 20 theorems, 129 equations)

This paper contains 14 sections, 20 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Outline of the Paper
  4. Preliminaries
  5. Notation
  6. Classical $BV$ Spaces
  7. Weighted $BV$ Spaces
  8. $A_1$ Weights
  9. A Structure Theorem for $BV_{\mathrm{loc}}(\Omega;w)$
  10. Sets of Finite $w$-Perimeter
  11. Smooth Approximation in $BV(\Omega;w)$
  12. Weighted Isoperimetric Inequalities
  13. Isometrically Embedding $BV(\Omega;w)\hookrightarrow BV(\Omega_w)$
  14. Characterization of $\mathcal{M}_F$

Key Result

Theorem 1.1

Let $w:\mathbb{R}^n\to(0,\infty]$ be lower semicontinuous, $f\in BV_\mathrm{loc}(\Omega;w)$. Then, there exist a Radon measure $\lVert Df\rVert_w$ and a $\lVert Df\rVert_w$-measurable function $\nu:\Omega\to\mathbb{R}^n$ such that In particular, $d\lVert Df\rVert_w=w\,d\lVert Df\rVert$.

Theorems & Definitions (56)

  • Theorem 1.1: Structure Theorem for $BV_{\mathrm{loc}}(\Omega;w)$
  • Theorem 1.2: Approximation by Smooth Functions
  • Theorem 1.3: Gagliardo-Nirenberg-Sobolev Inequality for $BV(\mathbb{R}^n;w)$
  • Remark 1.4
  • Corollary 1.5: Global Weighted Isoperimetric Inequality
  • Theorem 1.6: Isometrically Embedding $BV(\Omega;w)\hookrightarrow BV(\Omega_w)$
  • Definition 2.1: EG
  • Theorem 2.2: EG, Structure Theorem for $BV_\mathrm{loc}(\Omega)$
  • Definition 2.3
  • Lemma 2.4: Relationship between Weighted and Unweighted $BV$ Spaces
  • ...and 46 more