Dimension-independent estimates for heat operators and harmonic functions

TL;DR

This paper provides dimension-independent estimates for heat operators and harmonic functions on manifolds, offering new bounds and contractivity results applicable to various geometric and algebraic settings.

Contribution

It introduces a general contractivity framework for Markov kernels and derives dimension-independent bounds for harmonic functions on manifolds and Lie groups.

Findings

Dimension-independent estimates for heat operators.

Pointwise bounds for harmonic and subharmonic functions.

Applicability to Riemannian manifolds and Lie groups with curvature conditions.

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.