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The interest in harmonic functions on graphs goes back to the
nineteenth century, closely related to electrical networks and random walks.
Discrete Potential Theory is nowadays an active field, with connections and
applications to different areas of pure and applied mathematics.
Motivated by the continuous p-laplacian, some discrete counterparts
have been considered, like the discrete p-laplacian and, more recently, the so
called game p-laplacian. In such cases, the corresponding solutions satisfy
a local mean value property of the type u(x) = F (u(x1 ), ..., u(xd )) where
x1 , ..., xd are the neighbours of x. F is called an averaging operator on the
Harnack and Liouville properties are also central topics in Discrete Po-
tential Theory. In the case of the discrete p-laplacian, the Harnack(and
therefore Liouville) property was established by Holopainen-Soardi (1997)
under certain geometrical assumptions on the graph. Their method is, how-
ever, quite indirect because it follows the continuous road, with a Cacciopoli-
type inequality, the De-Giorgi-Moser iteration method and a discrete version
of the John-Nirenberg lemma as the key ingredients. Holopainen and Soardi
suggested the convenience of a more direct argument, only based on the local
In the talk we will report an elementary compactness proof of Liouville
theorem for averaging operators on the grid Zd , including the cases of the
discrete and game p-laplacians. (Joint work with T. Adamowicz).