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Characterizations of Sobolev and BV functions on Carnot groups

Francesco Serra Cassano, Kilian Zambanini

Abstract

We establish two characterizations of real-valued Sobolev and BV functions on Carnot groups. The first is obtained via a nonlocal approximation of the distributional horizontal gradient, while the second is based on an $L^p$ Taylor approximation, in the spirit of the results by Bourgain, Brezis and Mironescu.

Characterizations of Sobolev and BV functions on Carnot groups

Abstract

We establish two characterizations of real-valued Sobolev and BV functions on Carnot groups. The first is obtained via a nonlocal approximation of the distributional horizontal gradient, while the second is based on an Taylor approximation, in the spirit of the results by Bourgain, Brezis and Mironescu.

Paper Structure

This paper contains 7 sections, 31 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Carnot groups
  4. Differential calculus on Carnot groups
  5. Convolution on Carnot groups
  6. Nonlocal approximations of the horizontal gradient
  7. Characterizations via Taylor approximation and some consequences

Key Result

Theorem 1.1

Let $\mathbb G$ be a Carnot group endowed with a homogeneous norm $N$. Let $(\rho_\varepsilon)_\varepsilon$ be a sequence of mollifiers satisfying assumptions $P1\divP4$, and let $f\in L^p(\mathbb G)$, with $1\leqslant p<\infty$.

Theorems & Definitions (76)

  • Theorem 1.1: Characterization of Sobolev and BV functions by nonlocal gradients
  • Theorem 1.2: Characterization of Sobolev and $BV$ functions by $L^p$-Taylor approximation
  • Definition 2.1: H-linear maps
  • Proposition 2.2
  • Definition 2.3: Pansu differentiability
  • Theorem 2.4: Pansu-Rademacher Theorem Pansu
  • Example 2.5: The Heisenberg group
  • Definition 2.6: Horizontal Sobolev space
  • Theorem 2.7
  • Definition 2.8: Functions of bounded variation
  • ...and 66 more