Capacities Characterizing Removable Sets for Various Function Spaces in Carnot Groups

TL;DR

This paper characterizes removable sets for various function spaces in Carnot groups using capacities linked to hypoelliptic differential operators, extending classical removability concepts to sub-Riemannian geometries.

Contribution

It introduces capacity-based criteria for removability in Carnot groups for multiple function spaces, generalizing previous Euclidean results to sub-Riemannian settings.

Findings

Removability characterized via capacities for Campanato, Hölder, and $L^p_{loc}$ functions.

Lipschitz removability characterized with respect to the sub-Laplacian.

Extends classical Euclidean removability results to Carnot groups.

Abstract

We study removable sets for the Campanato, H\"{o}lder 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.