  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.
