  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 $f$.
