Local Second Order Sobolev Regularity for p-Laplacian Equation in Semi-Simple Lie Group

In this paper, we establish a structural inequality of the infinity-subLaplacian Delta(0,infinity )in a class of the semi-simple Lie group endowed with the horizontal vector fields X-1,. . .,X-2n. When 1 < p <= 4 with n = 1 and 1 < p < 3 + 1/n-1 with n >= 2, we apply the structural inequality to obtain the local horizontal W-2,W- 2-regularity of weak solutions top-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces R(2n )with n >= 2, the range of this p obtained is already optimal.