| In recent years, there has been considerable development in the study of Hardy type inequalities by making use of the method and tool of harmonic analysis, and it turns out that these inequalities play important roles in many branches of mathematics and physics. In order to meet the needs of the development of mathematics and the needs of solving some related problems in physics, it is necessary to study and improve Hardy inequality in all directions.In this paper, we will make a systematic and deep research on the theories and methods of Hardy inequality and Rellich inequality by reading some latest papers and materials, using and developing some useful skills in harmonic analysis and partial differential equation. I hope to get rich results in this research area. The main content of this paper is as follow.(1) In2009, Kombe and Ozaydin study some Hardy inequalities on a class of Riemannian manifolds. Particularly, they obtained some improved L2-Hardy inequalities on Hyperbolic spaces with sharp constants. In the first part of this paper, I will prove some Hardy-Rellich inequalities on Riemannian manifolds by a more direct and simple method, especially, I extend the L2-Hardy inequalities to the L2-case, moreover, I get the sharp constants in some cases. (2) In2010, Avkhadiev and Laptev obtained some improved Hardy inequalities on a class of nonconvex domains in R". In the second part of this paper, I prove a similar result on a class of nonconvex domains on the Heisenberg group.(3) Lewis etc got some improved Hardy inequalities on mean convex domains in2012, which extend some known results on convex domains. In the final part, I will firstly research a class of weighted Hardy inequality on mean convex domains, then prove a L2-Rellich inequality on mean convex domains by the help of the weighted Hardy inequality.In this paper, I obtain some new Hardy-Rellich type inequalities by using some new approaches. Especially, there are only few results known for the study on nonconvex domains and mean convex domains. I will study Hardy inequality more systematically and deeply in the future. |
