Hardy's Inequations And Properties Of Integrable Functions On NA Groups

Having in mind similar problems in the Euclidean and H-type groups setting, first we generalized Hardy-Sobolev inequalities from smooth functions with compact support to radial derivatives in Euclidean spaces, and point out that the constants in such Hardy-Sobolev inequalities are the best. Secondly, we prove Hardy-Rellich inequalities on Canrot groups as the same as on Heisenberg groups, and the constants in these inequalities are sharp. We show the analogue inequalities for Grushin operators also.NA groups include some Riemannian symmetric spaces with rank one as com-plex hyperbolic spaces, quaternionic hyperbolic spaces and Cayley hyperbolic space. We give a sufficient condition for integrable functions satisfy mean value property on NA groups to be harmonic with spherical transformations and spectral synthe-sis on analytic functions in strip.The method comes from the similar result about Riemannian symmetric spaces with rank one.H-type groups as nilpotent manifolds and NA groups as solvable manifolds are important Riemann manifolds. We obtain the holonomy groups of H-type groups and NA groups according to Berger classification of Riemannian holonomy groups. We use the spherical transformation to prove the semisimplity of radial group alge-bras on NA groups, and show these algebras are amenable.