Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

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