This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space M × X, where M is a (weighted) Riemannian manifold and X is a countable (symmetrically weighted) graph. Since some standard arguments for the elliptic case fail in this "mixed" setting, we adapt ideas from the discrete parabolic case found in Delmotte, 1999. We then present some useful applications of this inequality, namely, a kernel estimate for functions of the Laplacian on a graph that are in the spirit of Cheeger-Gromov-Taylor, 1982. This application in turn provides sharp estimates for certain Markov kernels on graphs, as suggested in Section 4 of a forthcoming paper by Persi Diaconis and the second author. We then close with an application to convolution power estimates on finitely generated groups of polynomial growth.

The first author's research was supported in part by NSF Grant DMS-0739164.
The second author's research was supported in part by NSF Grant DMS-1604771.