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arXiv · 1208.4614

Dimension-independent estimates for heat operators and harmonic functions

Brian C. Hall·Matthew Cecil
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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.

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Brian C. Hall, Matthew Cecil. 2012-08-22. Dimension-independent estimates for heat operators and harmonic functions. https://arxiv.org/abs/1208.4614

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