On potentials for sub-Laplacians and geometric applications

TL;DR

This paper extends potential theory to homogeneous Carnot groups, providing new estimates for polar sets of potentials and applying these results to various CR geometries.

Contribution

It introduces a novel approach leveraging geometric completeness and inequalities to estimate Hausdorff dimensions of polar sets in Carnot groups, expanding applications in CR geometries.

Findings

Establishment of inequalities analogous to Riesz potentials in Carnot groups

New estimates for Hausdorff dimensions of polar sets of potentials

Applications to CR, quaternionic CR, and octonionic CR geometries

Abstract

In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry.