Asymptotically mean value harmonic functions in sub-Riemannian and RCD settings

TL;DR

This paper explores the relationship between asymptotically mean value harmonic functions and classical harmonic functions across sub-Riemannian, RCD, and other geometric settings, establishing new characterizations and theorems.

Contribution

It demonstrates that Cheeger harmonic functions are weakly amv-harmonic in non-collapsed RCD spaces and characterizes harmonicity in Carnot groups via amv-harmonicity, including a Blaschke-Privaloff-Zaremba type theorem.

Findings

Cheeger harmonic functions are weakly amv-harmonic in non-collapsed RCD spaces.

Weak amv-harmonicity characterizes harmonicity in Carnot groups.

A Blaschke-Privaloff-Zaremba type theorem is proved for Carnot groups of step 2.

Abstract

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step $2$, we prove a Blaschke-Privaloff-Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.