# Regularity of inhomogeneous Quasi-linear equations on the Heisenberg Group

## TL;DR

This paper proves that weak solutions to certain quasi-linear equations in the Heisenberg Group have a horizontally continuous gradient, advancing understanding of regularity in non-Euclidean geometric contexts.

## Contribution

It establishes Holder continuity of the horizontal gradient for inhomogeneous quasi-linear p-Laplacian equations on the Heisenberg Group, a novel regularity result.

## Key findings

- Horizontal gradient of solutions is Holder continuous.
- Extends regularity theory to non-homogeneous equations in sub-Riemannian geometry.
- Provides new tools for analysis on the Heisenberg Group.

## Abstract

We establish Holder continuity of the horizontal gradient of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg Group.