  We establish dimension-independent estimates related to heat operators e^{tL}
on manifolds. We first develop a very general contractivity result for Markov
kernels which can be applied to diffusion semigroups. Second, we develop
estimates on the norm behavior of harmonic and non-negative subharmonic
functions. We apply these results to two examples of interest: when L is the
Laplace--Beltrami operator on a Riemannian manifold with Ricci curvature
bounded from below, and when L is an invariant subelliptic operator of
H\"ormander type on a Lie group. In the former example, we also obtain
pointwise bounds on harmonic and subharmonic functions, while in the latter
example, we obtain pointwise bounds on harmonic functions when a generalized
curvature-dimension inequality is satisfied.
