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

TL;DR

This paper develops a method to estimate the resolvent kernel of the Laplacian on p.c.f. self-similar fractals, providing decay bounds and a series decomposition useful for analyzing heat kernels and fractal blowups.

Contribution

It introduces a self-similar series decomposition approach for the Laplacian resolvent on fractals, enabling new estimates and applications.

Findings

Established upper bounds for the resolvent kernel on fractals.

Proved decay estimates on the positive real axis.

Demonstrated the existence of a self-similar series decomposition for the resolvent.

Abstract

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.