Asymptotic Mean Value Formula And L~p Hardy Type Inequality In The H-type Group

The paper is about the asymptotic mean value formula, the Hardy type inequality and the pseudo-differential operator on the H-type group.In chapter 1, we introduce some basic knowledge and our main results.We characterize the relationship between the subelliptic p-Laplace equation and the asymptotic expansion in the H-type group in chapter 2. A fundamental solution for the Kohn-Laplacian and the mean value formula for the Kohn-Laplacian in the H-type group are studied.In chapter 3, we show the Hardy-type inequalities in the H-type group. We also show the Hardy type inequalities in the half-space on the H-type group with the Green’s function. The LpHardy type inequalities in the H-type group and the halfspace on the H-type group are also given.Chapter 4 shows the pseudo-differential operators on the H-type group. We present the trace formula for the a-Weyl transform. Also, we give the characterization of Hilbert-Schmidt class and trace class pseudo-differential operators in the H-type group. We give necessary and sufficient conditions such that pseudo-differential operators on the H-type group are in the Hilbert-Schmidt class. A trace formula for trace class pseudo-differential operators in the H-type group is presented.