Archiv der Mathematik

, Volume 80, Issue 5, pp 516–524 | Cite as

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

Original paper


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 (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.

Mathematics Subject Classification (2000):

31B05 35B05 35J70 43A80 


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Copyright information

© Birkhäuser-Verlag 2003

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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