The Liouville Theorem for Discrete Symmetric Averaging Operators

We introduce averaging operators on lattices \(\mathbb Z^d\) ${\mathbb{Z}}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, p-harmonic, \(\infty \) $\infty$ -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 \(\mathbb Z^d\) ${\mathbb{Z}}^d$ .