Four Yuan On The Heisenberg Group Average Theorem And The Only Extension,

In this thesis, we study following three parts on quaternionic Heisenberg groups:First of all, in the view of the mean value theorem and Hardy inequalities playing an important role in partial differential equations and related subjects on the Euclid space, we establish the mean value theorem for the sub-Laplacian on quaternionic Heisenberg groups. As some applications, we obtain Hardy inequalities and the uncertainty principle by using the mean value theorem in the case of p=2. What's more, by using the method of Picone's identity, we get the Hardy inequalities for general p on quaternionic Heisenberg groups.Secondly, by techniques of surveying spherical functions, we investigate unique continuation properties on quaternionic Heisenberg groups, and obtain some interesting results.At last, we establish the fundamental solution of the p-sub-laplacian on quaternionic Heisenberg groups and prove an interesting facts that quaternionic Heisenberg groups are polarizable Carnot groups.