  We prove a global Li-Yau inequality for a general Markov semigroup under a
curvature-dimension condition. This inequality is stronger than all classical
Li-Yau type inequalities known to us. On a Riemannian manifold, it is
equivalent to a new parabolic Harnack inequality, both in negative and positive
curvature, giving new subsequents bounds on the heat kernel of the semigroup.
Under positive curvature we moreover reach ultracontractive bounds by a direct
and robust method.
