In this paper we study the inversion map and the Kelvin transform on the Heisenberg group ${\mathbb{H}}^n$. We first analyze the invariance of the Kelvin transform and provide an algebraic proof to the formula involving the sub-Laplacian. Furthermore, we apply the formula to seek the cylindrically symmetric solution to a sub-elliptic equation on ${\mathbb{H}}^n$ and determine the best constant of the Hardy-Sobolev type inequality.

In this article, we investigate the regularities of solutions to a class of nonlinear elliptic equations with measure data. These equations involve the $N$-functions, and the solutions belong to the Sobolev-Orlicz spaces. Through the application of comparison arguments, Caccioppoli-type inequality, and maximal estimate, we derive pointwise Riesz potential estimates for both the gradient of the solutions and the solutions themselves. Furthermore, we establish Hölder continuity estimates for the solutions.