Weighted Hardy's And Rellich's Inequalities On Heisenberg Groups

This thesis is devote to studying the weighted Hardy inequality,Rellich inequality,and the weighted Hardy and Rellich inequality with remainder terms on the Heisenberg group.The Heisenberg group is a typical representative of nilpotent Lie groups.Its research provides important theoretical basis for nilpotent Lie groups.Due to the complexity of the group structure and difference from Euclidean space.The theory of Heisenberg group is more challenge and significance.Our results the theoretical research of and will play an important role in the theory research to the nilpotent Lie groups and partial differential equations.Therefore,this research has both important scientific significance and research value.The thesis is organized as follows.Chapter 1,we present the background and significance of Hardy and Rellich inequalities.It is shown research progress of Hardy inequality and Rellich inequalities on Euclidean space and Heisenberg group.Moreover,the basic knowledge of the Heisenberg group is introduced.Chapter 2,the weighted Hardy inequality and Rellich inequality were proved,and the best constants of the weighted Hardy inequality are discussed,as well as the best constants of weighted Rellich inequality with a special cases.Then,the higher-order weighted Hardy-Rellich inequality is deduced by the iteration.Chapter 3,we prove the weighted Hardy and Rellich inequalities with the remainder term,and discuss the best constants of the weighted Hardy inequality with the remainder term.Moreover,by the iteration we obtain higher-order weighted Hardy-Rellich inequalities with remainder terms.