Heat kernel on smooth metric measure spaces and applications (Q289841)

Given a smooth function \(f\) on a complete Riemannian manifold, the authors derive a Harnack inequality for positive solutions of the \(f\)-heat equation \[ (\partial_t-\Delta_f)u=0,\quad \Delta_f=\Delta-\nabla f\cdot \nabla, \] and Gaussian upper and lower bound estimates for the \(f\)-heat kernel on complete smooth metric measure spaces with Bakry-Émery Ricci curvature bounded below. It is proved that both upper and lower bound estimates are sharp when the Bakry-Émery Ricci curvature is nonnegative. The machinery used in the proofs rely on the De Giorgi-Nash-Moser theory. As applications, the authors prove an \(L^1_f\)-Liouville theorem for \(f\)-subharmonic functions and an \(L^1_f\)-uniqueness theorem for \(f\)-heat equations when \(f\) has at most linear growth. Eigenvalues estimates and \(f\)-Green's function estimates are also obtained for the \(f\)-Laplace operator \(\Delta_f\).