Metric Growth of the Discrete $q$-Laplacian

Abstract

The metric growth of Laplacian $\Delta u$ in the theory of potentials on Riemannian manifolds by [13] means the growth of Green potential of $(\Delta u)^2$. This notion was introduced to obtain a Riesz representation for a biharmonic functions. On a hyperbolic network, we investigated a similar problem in [7], by taking the discrete Laplacian $\Delta$ and the (discrete) Green function. Let $q$ be a non-negative function such that $q \not\equiv 0$. Then $- \Delta + q$ is a Schrödinger operator. Let us put $\Delta _q := \Delta - q$ and call it $q$-Laplacian. The $q$-Green function $g_a$ of the network with pole at $a$ always exists. In this paper, we estimate the metric growth of $\Delta _q u$ by $q$-Green potential of $|\Delta _q u|$ and $|\Delta _q u|^2$. As an application, we prove Riesz representation theorem for $q$-biharmonic functions.