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Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups


Author: Luke G. Rogers
Journal: Trans. Amer. Math. Soc. 364 (2012), 1633-1685
MSC (2010): Primary 28A80, 60J35
Published electronically: October 24, 2011
MathSciNet review: 2869187
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Abstract:

A central issue in studying resistance forms on post-critically finite self-similar sets is the behavior of the resolvent of the Laplacian operator. The kernel of this operator may be obtained by a self-similar series and estimated on the right half of the complex plane via probabilistic bounds for the associated heat kernel. This paper generalizes the known upper estimates to the complement of the negative real axis in the complex plane via a new method using resistance forms and the Phragmen-Lindelöf theorem. Consequences include a proof of the self-similar structure of the resolvent on blow-ups, estimates for other spectral operators, and a new proof of the known upper estimates for the heat kernel on these sets.


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Additional Information

Luke G. Rogers
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: rogers@math.uconn.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05551-0
Received by editor(s): September 1, 2010
Received by editor(s) in revised form: January 27, 2011
Published electronically: October 24, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.