Differential Harnack estimates for conjugate heat equation under the Ricci flow

We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "Delta-R(x, t)", where R is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li-Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.