  In the first part, we derive a sharp gradient estimate for the log of
Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local
Li-Yau gradient estimate that matches the global one.
  In the second part, without explicit curvature assumptions, we prove a global
upper bound for the fundamental solution of an equation introduced by G.
Perelman, i.e. the heat equation of the conformal Laplacian under backward
Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a
qualitatively sharp, global Gaussian upper bound.
