Kelvin transform on the Heisenberg group revisited and applications to the best constant of Hardy-Sobolev type inequality

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.