Asymptotically mean value harmonic functions in doubling metric measure spaces

TL;DR

This paper extends the concept of harmonic functions characterized by mean value properties to doubling metric measure spaces, establishing regularity results and PDE connections in this generalized setting.

Contribution

It introduces and analyzes asymptotic mean value harmonic functions in doubling metric measure spaces, linking them to fractional Sobolev spaces and PDEs.

Findings

Strong amv-harmonic functions are Hölder continuous below exponent one.

Functions with finite amv-norm belong to fractional Hajlasz-Sobolev spaces.

In weighted Euclidean spaces, amv-harmonic functions satisfy a specific elliptic PDE.

Abstract

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are H\"older continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajlasz-Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.