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Capacities Characterizing Removable Sets for Various Function Spaces in Carnot Groups

Zack Boone

TL;DR

The paper extends the classical capacity-removability theory to the sub-Riemannian setting of Carnot groups. By developing Carnot-specific capacities κ, κ^{ρ}, κ^{δ}, and κ^{p} and establishing their sharp ties to Hausdorff content, it characterizes removability for Lipschitz, Campanato, Hölder, and L^p_loc function spaces via zero-capacity criteria. The approach blends fundamental solutions for left-invariant homogeneous operators with a Carnot-adapted Harvey-Polking partition of unity, overcoming noncommutativity to obtain robust geometric-measure criteria. These results generalize Euclidean removability theorems to the Carnot context and provide precise, quantitative conditions for when a set carries no information for a broad class of subelliptic PDEs. The framework has potential implications for geometric measure theory and analysis on Lie groups in sub-Riemannian geometries.

Abstract

We study removable sets for the Campanato, Hölder continuous, $L^p_{\text{loc}}$, and Lipschitz functions in Carnot groups. In the former three cases, we characterize removability through the use of capacities with respect to any left-invariant linear differential operator $\mathcal{L}$ for which $\mathcal{L}$ and $\mathcal{L}^t$ are hypoelliptic and satisfy a homogeneity condition, while in the latter case we characterize Lipschitz functions with respect to the sub-Laplacian.

Capacities Characterizing Removable Sets for Various Function Spaces in Carnot Groups

TL;DR

The paper extends the classical capacity-removability theory to the sub-Riemannian setting of Carnot groups. By developing Carnot-specific capacities κ, κ^{ρ}, κ^{δ}, and κ^{p} and establishing their sharp ties to Hausdorff content, it characterizes removability for Lipschitz, Campanato, Hölder, and L^p_loc function spaces via zero-capacity criteria. The approach blends fundamental solutions for left-invariant homogeneous operators with a Carnot-adapted Harvey-Polking partition of unity, overcoming noncommutativity to obtain robust geometric-measure criteria. These results generalize Euclidean removability theorems to the Carnot context and provide precise, quantitative conditions for when a set carries no information for a broad class of subelliptic PDEs. The framework has potential implications for geometric measure theory and analysis on Lie groups in sub-Riemannian geometries.

Abstract

We study removable sets for the Campanato, Hölder continuous, , and Lipschitz functions in Carnot groups. In the former three cases, we characterize removability through the use of capacities with respect to any left-invariant linear differential operator for which and are hypoelliptic and satisfy a homogeneity condition, while in the latter case we characterize Lipschitz functions with respect to the sub-Laplacian.

Paper Structure

This paper contains 9 sections, 30 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Carnot groups
  4. Left-invariant homogeneous operators and fundamental solutions.
  5. Capacity and Removability for Lipschitz Harmonic Functions
  6. Dyadic tilings and the Harvey-Polking Lemma in $\mathbb{G}$
  7. Capacity and Removability for Campanato Spaces
  8. Capacity and Removability for Hölder Continuous Functions
  9. Capacity and Removability for $L^p_{\text{loc}}$

Key Result

Theorem 1.1

Let $K \subset \mathbb{G}$ be bounded and $\mathcal{L}$ the sub-Laplacian. Then $K$ is removable for Lipschitz $\mathcal{L}$-solutions if and only if $\kappa(K) = 0$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 52 more