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Primitives of volume forms in Carnot groups

Annalisa Baldi, Bruno Franchi, Pierre Pansu

Abstract

In the Euclidean space it is known that a function $f\in L^2$ of a ball, with vanishing average,is the divergence of a vector field $F\in L^2$ with$$\| F\|\_{ L^2(B)} \le C \|f\|\_{L^2(B)}.$$In this Note we prove a similar result in any Carnot group $\mathbb{G}$ for a vanishing average $f\in L^p$, $1\le p < Q$, where $Q$ is the so-called homogeneous dimension of $\mathbb{G}$.

Primitives of volume forms in Carnot groups

Abstract

In the Euclidean space it is known that a function of a ball, with vanishing average,is the divergence of a vector field withIn this Note we prove a similar result in any Carnot group for a vanishing average , , where is the so-called homogeneous dimension of .

Paper Structure

This paper contains 6 sections, 18 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results and Notations
  3. Main result
  4. Rumin's Complex
  5. Equivalent formulation and proof of Theorem \ref{['main result']}
  6. Appendix A: kernels in Carnot groups

Key Result

Theorem 2.2

Let $U \subset \mathbb G$ be an open set, $1 \le p \le \infty$, and $m \in \mathbb{N}$. Then: In addition, if $p < \infty$, the following hold:

Theorems & Definitions (38)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4: see FSSC_step2
  • Theorem 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4: see BFTT, Remark 2.6
  • Lemma 3.5
  • proof
  • ...and 28 more