On the $C^{1,\alpha}$ Regularity of $p$-Harmonic Functions in the Heisenberg Group

TL;DR

This paper proves that weak solutions to the p-Laplace equation in the Heisenberg group have horizontally derivatives that are locally Hölder continuous when p > 4, advancing understanding of regularity in sub-Riemannian geometry.

Contribution

It establishes the $C^{1,eta}$ regularity of horizontal derivatives for p-harmonic functions in the Heisenberg group, a significant step in sub-Riemannian regularity theory.

Findings

Horizontal derivatives are locally Hölder continuous for p > 4.

Provides a new regularity proof in the sub-Riemannian setting.

Enhances understanding of p-harmonic functions in the Heisenberg group.

Abstract

We present a proof of the local H\"older regularity of the horizontal derivatives of weak solutions to the $p$-Laplace equation in the Heisenberg group $\mathbb{H}^1$ for $p>4 $.