标题： 负曲率Hadamard流形上的Poisson核估计 显示英文标题

标题： Estimates of the Poisson kernel on negatively curved Hadamard manifolds

摘要： 设$M$为一个具有夹紧负曲率的$n$维Hadamard流形$-b^2 \leq K_M \leq -a^2$。 $M$在无穷远处的Dirichlet问题的解导致构造了一族相互绝对连续的概率测度$\{\mu_x\}_{x \in M}$，称为调和测度。 以基点$o \in M$为基准，$M$的泊松核是函数$P : M \times \partial M \to (0, \infty)$，由\begin{equation*} P(x, \xi) = \frac{d\mu_x}{d\mu_o}(\xi) \ , \ x \in M, \xi \in \partial M. \end{equation*}定义。 我们证明了泊松核的全局上界和下界如下： \begin{equation*} \frac{1}{C}\: e^{-2K{(o|\xi)}_x}\: e^{a d(x, o)} \le P(x,\xi) \le C\: e^{2K{(x|\xi)}_o}\: e^{-a d(x,o)} \:, \end{equation*}，其中正常数$C \geq 1, K > 0$仅依赖于$a, b$和$n$。 上述估计可以看作是Gromov双曲调和流形特殊情况下的泊松核的经典公式的推广。这些估计并不能直接从格林函数或调和测度的已知估计中得出。相反，我们使用了Anderson-Schoen的方法来估计锥体中的正调和函数。作为应用，我们得到了$\mu_x \to \delta_{\xi}$当$x \in M \to \xi \in \partial M$时的收敛性的定量估计，以及有限球上的调和测度趋近于无限边界上的调和测度的收敛性估计，当球的半径趋于无穷大时。 显示英文摘要

摘要： Let $M$ be an $n$-dimensional Hadamard manifold of pinched negative curvature $-b^2 \leq K_M \leq -a^2$. The solution of the Dirichlet problem at infinity for $M$ leads to the construction of a family of mutually absolutely continuous probability measures $\{\mu_x\}_{x \in M}$ called the harmonic measures. Fixing a basepoint $o \in M$, the Poisson kernel of $M$ is the function $P : M \times \partial M \to (0, \infty)$ defined by \begin{equation*} P(x, \xi) = \frac{d\mu_x}{d\mu_o}(\xi) \ , \ x \in M, \xi \in \partial M. \end{equation*} We prove the following global upper and lower bounds for the Poisson kernel: \begin{equation*} \frac{1}{C}\: e^{-2K{(o|\xi)}_x}\: e^{a d(x, o)} \le P(x,\xi) \le C\: e^{2K{(x|\xi)}_o}\: e^{-a d(x,o)} \:, \end{equation*} for some positive constants $C \geq 1, K > 0$ depending solely on $a, b$ and $n$. The above estimates may be viewed as a generalization of the well-known formula for the Poisson kernel in terms of Busemann functions for the special case of Gromov hyperbolic harmonic manifolds. These estimates do not follow directly from known estimates on Green's functions or harmonic measures. Instead we use techniques due to Anderson-Schoen for estimating positive harmonic functions in cones. As applications, we obtain quantitative estimates for the convergence $\mu_x \to \delta_{\xi}$ as $x \in M \to \xi \in \partial M$, and for the convergence of harmonic measures on finite spheres to the harmonic measures on the boundary at infinity as the radius of the spheres tends to infinity.