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The Liouville theorem for discrete symmetric averaging operators (2204.12295v2)

Published 26 Apr 2022 in math.AP and math.MG

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

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