有非负Ricci曲率的度量测度空间上调和函数的边界问题  

Boundary Behavior of Harmonic Functions on Metric Measure Spaces with Non-negative Ricci Curvature

作　　者：杨婉婉 李波 YANG Wanwan;LI Bo(Center for Applied Mathematics,Tianjin University,Tianjin,300072,P.R.China)

机构地区：[1]天津大学应用数学中心,天津300072

出　　处：《数学进展》2021年第2期245-258,共14页Advances in Mathematics（China）

基　　金：国家自然科学基金(Nos.11922114,11671039,11771043)。

摘　　要：设(X,d,μ)是满足非负Ricci曲率条件的度量测度空间.本文研究了(开)上半空间X×R+上调和函数的边界问题.我们得到了:若u(x,t)是定义在上半空间X×R+上的调和函数,且满足Carleson测度条件sup xB rB∫rB0fB(xB,xB)|t▽u(x,t)|^(2)dμ(x)dt/t≤C<∞,其中▽=(▽_(x),■t)表示全梯度且B(xB,rB)表示以xB为球心、rB为半径的(开)球,则它的迹u(x,0)=f(x)是有界平均振动(BMO)函数.反之,迹满足BMO条件的所有调和函数满足以上Carleson测度条件.Let(X,d,μ)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 halfspace X x R+.We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X x R+,u(x,0)=f(x),whenever u satisfies the following Carleson measure condition sup xB rB∫rB0fB(xB,xB)|t▽u(x,t)|^(2)dμ(x)dt/t≤C<∞where▽=(▽x,■t)denotes the total gradient and B(xB,rB)denotes the(open)ball centered at xB with radius rB.Conversely,the above condition characterizes all the harmonic functions whose traces are in BMO space.

关 键 词：调和函数 度量测度空间 有界平均振动 CARLESON测度 

分 类 号：O174.3[理学—数学]