Dimension-independent estimates for heat operators and harmonic functions
Abstract
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.
Cite
@article{arxiv.1208.4614,
title = {Dimension-independent estimates for heat operators and harmonic functions},
author = {Brian C. Hall and Matthew Cecil},
journal= {arXiv preprint arXiv:1208.4614},
year = {2014}
}
Comments
Corrected minor errors. Specifically, certain results that were claimed to hold for p greater than or equal to one are established only when p > 1