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

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.

Keywords

sub-Gaussian heat kernel estimates, resistance, Poincaré inequality

2010 Mathematics Subject Classification

35K08, 60J10, 60Jxx

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Dedicated to the memory of Professor Ka-Sing Lau

Research partially supported by NSERC (Canada)

Received 31 December 2022

Accepted 21 June 2023

Published 7 August 2024