A generalization of Hamilton’s gradient estimate

In this short note, we obtain a “ΦΦ-gradient estimate” for bounded solutions to the heat equation on compact Riemannian manifolds, which generalizes the well-known Hamilton gradient estimate [R.S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993) 113–126]. We derive as applications some useful estimates for the ΦΦ-entropy and the associated heat kernel.