Tohoku Mathematical Journal
2011
September
SECOND SERIES VOL. 63, NO. 3

Tohoku Math. J.
63 (2011), 303-327

Title PARABOLIC HARNACK INEQUALITY ON METRIC SPACES WITH A GENERALIZED VOLUME PROPERTY

Author Giacomo De Leva

(Received August 19, 2009, revised December 20, 2010)
Abstract. We study the parabolic Harnack inequality on metric measure spaces with the more general volume growth property than the volume doubling property. As applications we extend some Liouville theorems and heat kernel estimates for Riemannian manifolds to Alexandrov spaces satisfying a volume comparison condition of Bishop-Gromov type.

2000 Mathematics Subject Classification. Primary 58J35; Secondary 53C21, 31C25.
Key words and phrases. Alexandrov space, infinitesimal Bishop-Gromov condition, parabolic Harnack inequality, heat kernel, harmonic functions, subharmonic functions.

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