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We consider weakly and strongly asymptotically mean value harmonic(amv-harmonic) functions on subriemannian and RCD settings. We demonstratethat, 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 thesub-Laplacian.  In homogeneous Carnot groups of step $2$, we prove aBlaschke-Privaloff-Zaremba type theorem. Similar results are discussed in thesettings of Riemannian manifolds and for Alexandrov surfaces.

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