Discrete/Continuous Elliptic Harnack Inequality and Kernel Estimates for Functions of the Laplacian on a Graph

This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space $M \times X$, where $M$ is a (weighted) Riemannian Manifold and $X$ is a countable graph. Since some standard arguments for the elliptic case fail in this "mixed" setting, we adapt ideas introduced by Thierry Delmotte for the discrete parabolic case. We then present some useful applications of this inequality, namely, a kernel estimate for functions of the Laplacian on a graph. This application in turn provides sharp estimates for certain Markov kernels on graphs. We then close this paper with an application to convolution power estimates on finitely generated groups of polynomial growth.