A note on heat kernel estimates, resistance bounds and Poincaré inequality

Abstract

Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincaré inequality, capacity upper bound, and a slow volume growth condition. An important feature of this work is that we do not assume elliptic Harnack inequality, cutoff Sobolev inequality, or exit time bounds.

17 pages. arXiv admin note: text overlap with arXiv:1803.11296; fixed some typos and added a few references