 

Mark Cerenzia and Laurent SaloffCoste
Discrete/continuous elliptic Harnack inequality and kernel estimates for functions of the Laplacian on a graph view print


Published: 
August 10, 2013 
Keywords: 
Convolutions, Harnack inequality, Markov kernels 
Subject: 
42A85, 35K25, 60F99 


Abstract
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 CheegerGromovTaylor, 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.


Acknowledgements
The first author's research was supported in part by NSF Grant DMS0739164.
The second author's research was supported in part by NSF Grant DMS1604771.


Author information
Mark Cerenzia:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
cerenzia@princeton.edu
Laurent SaloffCoste:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
lsc@math.cornell.edu

