Boundary Value Problems Of Harmonic Functions On A Class Of Metric Space

Let(X,d,?)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X × R_+.We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X × R_+,u(x,0)= f(x),whenever u satisfies the following Carleson measure condition where B(x_B,r_B)represents a sphere with x_Bas its center and r_Bas its radius.? =(?_x,(?)_t)denotes the total gradient.Conversely,the condition(*)characterizes all the harmonic functions whose traces are in BMO space.The paper is mainly divided into four chapters:The first chapter is the introduction of the whole paper.This chapter mainly introduces the background and development status of the boundary value problem of the harmonic function on the upper half-space X × R_+,as well as the problems and results obtained in this paper.In the second chapter,basic concepts and relative results are presented.This chapter is divided into three sections.The first section introduces the Sobolev space on the metric space.The second Section introduces differential construction and Laplace operator,as well as the introduction of related function spaces.The third Section introduces the curvature dimension conditions and related lemmas,They are important tools to prove the main conclusions of this paper.In the third chapter,We obtain the relevant properties of Poisson kernels with the help of heat kernel,and explore the relationship between the harmonic functions on the upper half-space X × R_+ and Poisson semigroups.In the fourth chapter,we study the equivalent characterization of HMO space on the metric measurement space with non-negative Ricci curvature.