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Intrinsic perimeter, compactness and Poincaré inequality for SBV functions in Carnot-Carathéodory spaces

Marco Di Marco

TL;DR

The paper develops sub-Riemannian analogues of classical SBV theory in equiregular Carnot–Carathéodory spaces by introducing an intrinsic perimeter measure for countably X-rectifiable sets and assuming property $\mathcal{R}$. It proves a compactness theorem for SBV$_X$ functions and a Poincaré inequality, expressing the derivative $D_Xu$ in terms of the approximate gradient and the jump contribution via the intrinsic perimeter $\mathcal{P}^X_{\mathcal{J}_u}$. The core ideas combine the intrinsic perimeter with a chain-rule-based compactness argument and an isoperimetric-based Poincaré framework, enabling rigorous control of both the absolutely continuous and jump parts. These results establish sub-Riemannian SBV theory with tools such as the coarea formula and CC-space isoperimetry, and pave the way for variational and geometric analysis in CC settings.

Abstract

By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincaré inequality for special functions with bounded variation in equiregular Carnot-Carathéodory spaces which satisfy an additional natural assumption, called property $\mathcal R$.

Intrinsic perimeter, compactness and Poincaré inequality for SBV functions in Carnot-Carathéodory spaces

TL;DR

The paper develops sub-Riemannian analogues of classical SBV theory in equiregular Carnot–Carathéodory spaces by introducing an intrinsic perimeter measure for countably X-rectifiable sets and assuming property . It proves a compactness theorem for SBV functions and a Poincaré inequality, expressing the derivative in terms of the approximate gradient and the jump contribution via the intrinsic perimeter . The core ideas combine the intrinsic perimeter with a chain-rule-based compactness argument and an isoperimetric-based Poincaré framework, enabling rigorous control of both the absolutely continuous and jump parts. These results establish sub-Riemannian SBV theory with tools such as the coarea formula and CC-space isoperimetry, and pave the way for variational and geometric analysis in CC settings.

Abstract

By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincaré inequality for special functions with bounded variation in equiregular Carnot-Carathéodory spaces which satisfy an additional natural assumption, called property .
Paper Structure (5 sections, 15 theorems, 87 equations)

This paper contains 5 sections, 15 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Intrinsic perimeter measure of countably X-rectifiable sets
  4. Compactness of SBVX functions
  5. Poincaré inequality for SBVX functions

Key Result

Proposition 1.3

Let $\Omega$ be an open subset of an equiregular Carnot-Carathéodory space $(\mathbb{R}^n,X)$ satisfying property $\mathcal{R}$. The following statements are equivalent: where $u^\pm$ are the traces of $u$ on $\mathcal{J}_u$ (see Definition def_approxjump) and $\nu_{\mathcal{J}_u}$ is the horizontal normal of $\mathcal{J}_u$ (see Definition def_ipersup). Observe that ${\mathcal{P}^X_{\mathcal{J}_

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 31 more