# Borderline gradient estimates at the boundary in Carnot groups

## TL;DR

This paper establishes boundary gradient continuity for solutions to perturbed horizontal Laplace equations in Carnot groups, extending regularity results to critical Lorentz space settings and non-characteristic boundary portions.

## Contribution

It proves boundary gradient continuity for solutions to Dini-perturbed horizontal Laplace equations in Carnot groups, advancing regularity theory in subelliptic PDEs.

## Key findings

- Proves continuity of horizontal gradient near non-characteristic boundary portions.
- Extends boundary regularity results to Lorentz space perturbations.
- Provides a subelliptic analogue of classical boundary regularity theorems.

## Abstract

In this article, we prove the continuity of the horizontal gradient near a $C^{1,\text{Dini}}$ non-characteristic portion of the boundary for solutions to $\Gamma^{0, \text{Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below where the scalar term is in scaling critical Lorentz space $L(Q,1)$ with $Q$ being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the $\Gamma^{1, \alpha}$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.