Hardy Theorem On Some Lie Group And The Hardy Uncertainty Principle

This thesis is finished under the guidance of my supervisor Professor Niu Pehcheng. In Chapter one, the theory of irreducible unitary representations on two-step nilpotent Lie groups is introduced. Then by using the Cowling and Price's idea that is to prove a version of Hardy's uncertainty principle in R" , we get an L~p- L~q version of Hardy's uncertainty principle on two-step nilpotent Lie groups. Furthermore, via the Morgan's method to obtain Hardy's uncertainty principle in R", we establish another version (Morgan's version) of Hardy's uncertainty principle on two-step nilpotent Lie groups. In Chapter two, at first, the properties of spherical functions and the Laplace-Beltrami operator's heat kernels on noncompact rank one symmetric spaces are described. Next, we set up an L~p - L~q version of Hardy's theorem on noncompact rank one symmetric spaces in terms of the Helgason-Fourier transforms. In Chapter three, the properties of Poisson kernel and spherical functions on harmonic NA groups are presented. Then, we give a version of Hardy's uncertainty principle on harmonic NA groups from the Helgason-Fourier transforms and the Laplace-Beltrami operator's heat kernels.