The weak elliptic Harnack inequality revisited

In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincaré inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.

Keywords

weak Harnack inequality, lemma of growth, mean exit time, Dirichlet form

2010 Mathematics Subject Classification

28A80, 30Lxx, 31C25, 35K08

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Dedicated to the memory of Professor Ka-Sing Lau

The first-named author was supported by the NationalNatural Science Foundation of China (No. 12271282).

Published 12 July 2024