The Liouville Theorem for Discrete Symmetric Averaging Operators

Abstract We introduce averaging operators on lattices $$\mathbb Z^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, p -harmonic, $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -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 $$\mathbb Z^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> .