On Some Function Spaces And Variational Problems Related To The Heisenberg Group

The main aim of this thesis is to study the properties of the maps for Heisenberg group target, which include Lipschitz and Holder continuity, L^P(#,Hⁿ) , W^1,p(#,Hⁿ) , BMO and John-Nirenberg estimates, embedded theorems, Poincare inequalities and reverse Poincare inequalities, the regul-arities about the minimizers. In the last part, we study the existence of the solutions for semilinear subelliptic equations and the estimates of the eigenvalues for subelliptic operators on H". The results are generalizations for the appropriate ones on R".In Chapter 1 we recall the definitions for the Heisenberg group H", state some properties in geometry and analysis and in Holder continuous maps for Heisenberg group targets. In Chapter 2, we study the properties of L^P(#,Hⁿ) and W^1,P(#,Hⁿ), the pre-norm and pre-weakly compactness on W^1,P(#,Hⁿ) BMO(#,Hⁿ) and John-Nirenberg type estimate. In Chapter 3, we state and prove some Poincare inequalities and reverse Poincare inequalities on the Heisenberg group. In chapter 4, we state and prove Campanato type Theorem, Morrey type Theorem and some embedding Theorems for the maps. In Chapter 5, we first recall some results of the existence and regularity about the minimizers. Secondly, we study the Euler equations about the variationals and the approximate problems about the minimizers. In chapter 6, we study existence of the weak solutions for a class of the nonlinear Dirichlet problems on H" and estimates of the eigenvalues about the subelliptic operators on Heisenberg group.