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Closed BV-extension and $W^{1,1}$-extension sets

Emanuele Caputo, Jesse Koivu, Danka Lučić, Tapio Rajala

TL;DR

This work analyzes when extension properties transfer between $BV$ and $W^{1,1}$ function classes from closed sets in metric measure spaces. Under doubling measures and a weak $(1,1)$-Poincaré inequality (PI spaces), the authors establish measure-density and decomposition results for extension sets, and relate homogeneous to full BV extensions. In the Euclidean plane, they prove that compact $BV$-extension sets with finitely many complementary components are $W^{1,1}$-extension sets, leveraging a local quasiconvexity of the complement and perimeter modification arguments. The findings clarify the landscape of extension properties, provide explicit positive results under geometric and analytic conditions, and supply counterexamples illustrating the limits of general implications.

Abstract

This paper studies the relations between extendability of different classes of Sobolev $W^{1,1}$ and $BV$ functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak $(1,1)$-Poincaré inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected $BV$-extension sets are always also $W^{1,1}$-extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.

Closed BV-extension and $W^{1,1}$-extension sets

TL;DR

This work analyzes when extension properties transfer between and function classes from closed sets in metric measure spaces. Under doubling measures and a weak -Poincaré inequality (PI spaces), the authors establish measure-density and decomposition results for extension sets, and relate homogeneous to full BV extensions. In the Euclidean plane, they prove that compact -extension sets with finitely many complementary components are -extension sets, leveraging a local quasiconvexity of the complement and perimeter modification arguments. The findings clarify the landscape of extension properties, provide explicit positive results under geometric and analytic conditions, and supply counterexamples illustrating the limits of general implications.

Abstract

This paper studies the relations between extendability of different classes of Sobolev and functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak -Poincaré inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected -extension sets are always also -extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.

Paper Structure

This paper contains 14 sections, 24 theorems, 192 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Spaces $BV$ and $W^{1,1}$
  5. PI spaces
  6. Extension sets in general metric measure spaces
  7. On the relations between $W^{1,1}$- and $W^{1,1}_w$-extension sets
  8. Extension sets in PI spaces
  9. Measure density
  10. From homogeneous norm to full norm
  11. Connectivity and decomposition properties of $BV$- and $\overset{\circ}{{BV}}$-extension sets
  12. Compact $BV$-extension sets in the Euclidean plane
  13. Quasiconvexity of the complement of extension sets
  14. From $BV$-extension to $W^{1,1}$-extension

Key Result

Theorem 1.1

Let $({\rm X},{\sf d},\mathfrak m)$ be a metric measure space. Let $\Omega \subset {\rm X}$ be a bounded open set. Then for every $\varepsilon >0$ there exists a closed set $G \subset \Omega$ such that $\mathfrak m(\Omega\setminus G) < \varepsilon$ and so that the zero extension gives a bounded oper

Figures (2)

  • Figure 1: The picture gives a qualitative description of the construction in Step 1. The region $A_1$ is enclosed by the dashed path $\gamma$ and the line segment $[x,y]$. The picture follows the construction in the case $A_1 \cap E \neq \emptyset$.
  • Figure 2: This is a qualitative description of the construction of $C_1$, $\alpha_1$ and $\beta_1$ for the example in Figure \ref{['fig:Modification_in_Omega']}. On the left-hand side, we have the set $C_1$, which in this case consists of two connected components, $C_1^1$ and $C^2_1$. Then, in the example, the image of the curve $\alpha_1$ is exactly $C_1^1$. Starting from $\alpha_1$, by making use of Lemma \ref{['lma:quasi_int']}, we build $\beta_1$ by replacing the four subcurves of $\alpha_1$ connecting the points $x$ to $x_1$, $x_1$ to $x_2$, $x_2$ to $x_3$, and $x_4$ to $y$ by the dashed curves connecting the same points, while the subcurve of $\alpha_1$ connecting $x_3$ to $x_4$ remains unchanged.

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.4
  • Proposition 1.5: Lemma \ref{['lma:quasiconvexity_full']}
  • Definition 2.1: Total variation
  • Definition 2.2: The spaces $\overset{\circ}{{BV}}(B)$ and $BV(B)$
  • Definition 2.3: Sets of finite perimeter on a Borel subset B
  • Definition 2.4
  • Definition 2.5: The spaces $L^{1,1}({\rm X})$ and $W^{1,1}({\rm X})$
  • Definition 2.6: The space $W^{1,1}(\Omega)$
  • Proposition 2.7: Smoothing operator CKR23
  • ...and 53 more