The Fundamental Solution to the p-Laplacian in a class of H\"ormander Vector Fields

TL;DR

This paper derives the fundamental solution to the p-Laplace equation within a specific class of H"ormander vector fields, extending understanding beyond Carnot groups and Grushin spaces, and applies it to harmonic functions and capacity calculations.

Contribution

It provides the first fundamental solution for the p-Laplace equation in a new class of H"ormander vector fields not associated with known structures.

Findings

Derived the fundamental solution at sub-Riemannian points.

Constructed an infinite harmonic function with prescribed boundary data.

Computed the capacity of annuli centered at the singularity.

Abstract

We find the fundamental solution to the p-Laplace equation in a class of H\"ormander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity.