Borderline gradient estimates at the boundary in Carnot groups

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 ${\mathrm{\Gamma }}^{0,\text{Dini}}$ 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 ${\mathrm{\Gamma }}^{1,\alpha }$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.