  We prove a generalization of the Li-Yau estimate for a board class of second
order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau
inequality and a new Harnack inequality for these equations. We also prove a
Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for
these equations. This results in a generalization of Huisken's monotonicity
formula for a family of evolving hypersurfaces. Finally, we also show that all
these generalizations are sharp in the sense that the inequalities become
equalities for a family of fundamental solutions, which however different from
the Gaussian heat kernels on which the equality was achieved in the classical
case.
