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

摘要

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_(r→0+) r~(-2)(M_ru-u) = 0 in the weak sense of distributions (M_r 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.