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

Rogers, Luke

doi:10.1090/S0002-9947-2011-05551-0

Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups
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Trans. Amer. Math. Soc. 364 (2012), 1633-1685 Request permission

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-Lindelof̈ 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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