A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow

We establish a point-wise gradient estimate for $all$ positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li-Yau estimate for the linear Schr\"odinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A generalization of Perelman's W-entropy is defined in both the Ricci flow and fixed metric case. We also find a new family of heat kernel estimates.
Comment: Some typos are removed from Corollary 1