A note on a generalized form of the Laplacian and of sub-Laplacians

In this note we prove a recent conjecture of Hasson [11]: we show that, for a locally integrable function u, a sufficient condition to be harmonic is that $ \lim\limits_{r\to 0^+} r^{-2}(M_{r}u-u) = 0 $ in the weak sense of distributions (Mr being the averaging operator on balls of radius r). We also extend this and other results to the setting of sub-Laplacians on Carnot groups.