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

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

2010-09-27

arxiv.org/abs/1009.5334v1

Mathematics

Classical Analysis and ODEs

46 pages

Scientific paper

We provide a method for obtaining upper estimates of the resolvent kernel of the Laplacian on a post-critically finite self-similar fractal that relies on a self-similar series decomposition of the resolvent. Decay estimates on the positive real axis are proved by analyzing functions satisfying an interior eigenfunction condition with positive eigenvalue. These lead to estimates on the complement of the negative real axis via the Phragmen-Lindelof theorem. Applications are given to kernels for functions of the Laplacian, including the heat kernel, and to proving the existence of a self-similar series decomposition for the Laplacian resolvent on fractal blowups.

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