Weighted Poincare Inequalities On Half Spaces In Carnot Groups

In recent years the topic of Hardy inequalities, Poincare inequalities and their applications seems to have become more and more popular. Although the original Hardy inequality, proved by G.H.Hardy, was discovered in the 1920’s, new versions are stated and proved and old ones are improved almost a century later. One reason for their popularity is their usefulness in various types of applications, for example, PDE,uncertainty principle et al. Later on, these inequalities have been generalized and modified in many different ways and the literature concerning such inequalities is extensive. Many other Hardy type inequalities may be found in the papers and books.In this paper,we shall consider the Hardy inequalities and Poincare inequalities in H-type groups, or nilpotent groups. The aim of the paper is to look for the sharp constants of such inequalities. If G is a H-type group, we shall show the sharp Hardy inequalities on half spaces for Kohn’s sublaplacian on G. As an application, we obtain some Rellich type inequalities for the same operator. If G is a Carnot group, we shall prove some weighted Poincare inequalities on half spaces for the sublaplacian in G. Furthermore, all the constants we obtain are sharp.