The Liouville Theorem for Discrete Symmetric Averaging Operators

We introduce averaging operators on lattices $$\mathbb Z^d$$ Z d and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, p -harmonic, $$\infty $$ ∞ -harmonic and the so-called game p -harmonic functions. Our approach provides an elementary alternative proof of the Liouville Theorem for positive p -harmonic functions on $$\mathbb Z^d$$ Z d .