Heisenberg Group Last Laplacian

In this thesis, we consider the eigenvalue for the subelliptic operator△_H and the gradient estimation of the fundamental solution for△_H-2.To obtain the existence and the estimation of the eigenvalue for the operator△_H, we use the variational method which is similar to the case in the Euclidean space.According to the estimation of the heat kernel for the operator△_H—(?)_s, we get the gradient estimation for the operator△_H-2. At the same time, we get the similar result for the non-divergence subelliptic operator sum from i,j to m a_i,jX_iX_j-2...