We prove a Yau’s type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry–Emery Ricci tensor and the weighted mean curvature are bounded below. As an application, we give a Liouville type result for bounded $f$-harmonic functions with the Dirichlet boundary condition. Our results do not depend on any assumption on the potential function $f$.

中文翻译：

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带 Dirichlet 边界条件的加权调和函数的梯度估计

我们证明了正的 Yau 类型梯度估计 $F$当无限维 Bakry-Emery Ricci 张量和加权平均曲率有界时，具有紧凑边界的平滑度量空间上的 Dirichlet 边界条件的 - 谐波函数。作为应用程序，我们给出了有界的 Liouville 类型结果$F$-具有狄利克雷边界条件的谐波函数。我们的结果不依赖于对潜在函数的任何假设$F$.