热方程在非负Ricci曲率度量测度空间上的Cauchy问题

Cauchy problems for heat equations in metric measure spaces with non-negative Ricci curvature

设(X,d,μ)是满足非负Ricci曲率条件的度量测度空间。在上半空间X×R_(+)上,考虑热方程的Cauchy问题。热方程为{∂_(x)u(x,t)-Δ_(x)u(x,t)=0,x∈X,t>0,u(x,0)=f(x),x∈X,其中Δ_(x)是X上的Laplace算子。我们得到了:若u(x,t)是热方程的解(称其为热函数)且满足Carleson测度条件(*)sup x_(B),r_(B) 1/μ(B(x_(B),r_(B)))∫_(0) ^(r_(B)^(2))∫_(B(x_(B),r_(B)))(|t∂_(t) u|+|√t∇_(x) u|)^(2) dμdt/t≤C,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条件的所有热函数u(x,t)恰好满足Carleson测度条件(*)式。

Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is focuses on the Cauchy problem for the heat equation on the upper half-space X×R_(+).The heat equation is{∂_(x)u(x,t)-Δ_(x)u(x,t)=0,x∈X,t>0,u(x,0)=f(x),x∈X,whereΔ_(x) is the Laplace operator on X.We derive that a function f of bounded mean oscillation BMO is the trace of solution u(x,t)of heat equation above(called Caloric function),u(x,0)=f(x),whenever u satisfies the following Carleson measure condition(*)sup x_(B),r_(B) 1/μ(B(x_(B),r_(B)))∫_(0) ^(r_(B)^(2))∫_(B(x_(B),r_(B)))(|t∂_(t) u|+|√t∇_(x) u|)^(2) dμdt/t≤C,Conversely,the condition(*)characterizes all the Caloric functions whose traces are in BMO space.