Some Hardy Type Inequalities In The Carnot Group

In paper we primarily research the expressions of the weighted Hardy typeinequalities, including the Heisenberg group on the Hardy type inequality, Heisen-berg type group on the Hardy type inequality and arbitrary step Carnot Group onthe Hardy type inequality,where the weight function w(K) and K is related to thefundamental solution of sub-Laplacian.The whole contents are organized as followsIn Chapter 1, we brie?y state the background of this thesis and the presentsituation on carnot group.The main results and methods are also presented.In Chapter 2, we state some relevant concepts of carnot group, heisenberggroup and heisenberg type group ,and prove some lemmas, which play an importantrole in the proof of weighted hardy type inequality in chapter 3.In Chapter 3, ?rst of all,we establish three kinds of hardy type inequalitieson the heisenberg group .Then, applied the method to solve the weighted hardyinequality on the heisenberg group,we extend the results to the heisenberg typegroup, and obtain three Hardy inequalities on the heisenberg type group.In Chapter 4, we put a chief conclusions to more general Carnot groups, onwhich we discuss its weighted hardy type inequality.