Multi-bit Some Research Of Potential Theory

In this thesis, we generalize the method of pluripotential theory for complex Monge-Ampere operator to that for the k-Hessian operator and quaternionic Monge-Ampere operator respectively. We get some interesting and completely new results, such as boundary measure, Lelong-Jensen type formula, Lelong num-ber, Green function, closed positive current and so on.In Chapter1, we give a comprehensive survey of the backgrounds and modern developments of complex Monge-Ampere operator, plurisubharmonic functions, closed positive current, k-Hessian measure and quaternionic Monge-Ampere operator. And then we introduce some idea and concepts referring to this thesis and the main results of the subject.In Chapter2, we mainly discuss the k-Hessian operator and k-convex func-tions. We show some basic properties and the weak convergence theorem for k-Hessian measure. Then by studying the relative extremal function, we establish an global approximation of negative k-convex functions on some k-hyperconvex domain. Moreover, we give several estimates for the mixed k-Hessian operator.Chapter3is devoted to the study of Lelong-Jensen type formula and Lelong number for k-convex functions. We find an explicit formula for k-Hessian bound-ary measure, and establish Lelong-Jensen type formula, which can be regarded as a k-convex version of Poisson integral formula. We also show the comparison theorem for k-Hessian boundary measure and introduce Lelong number and the generalized Lelong number for k-convex functions.In Chapter4, we study the k-Green functions with single pole and with several poles respectively. We prove their continuity by using different methods, and show that the the k-Green function is the unique solution to the Dirich-let problem. Their behavior on the boundary of k-hyperconvex domain is also studied.In the last chapter, we mainly discuss closed positive current on quaternionic space and the quaternionic Monge-Ampere operator. We show some properties and an explicit formula for Baston operator Δ. We say a current T is close if DT=0, where D is the second operator in the O-Cauchy-Fueter complex. Then we define Δu1∧...∧Δuk∧T as a closed positive current also in the case when the plurisubharmonic functions u1,..., uk are not bounded below, and prove the weak convergence theorem. Finally, we establish Lelong-Jensen type formula and Lelong number for quaternionic plurisubharmonic functions.