Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups (Q2880692)

The author studies the analysis of post-critically finite self-similar fractal sets. The best known example of a set of this type is the Sierpinski gasket. The paper continues investigations due to \textit{M. Ionescu} et al. [Trans. Am. Math. Soc. 362, No. 8, 4451--4479 (2010; Zbl 1204.28013)]. As was shown in this paper, the resolvent kernel \((z-\Delta)^{-1}\) of the Laplacian on that sets can be written as a self-similar series for suitable values of \(z\). The purpose of the present paper is to establish estimates that permit the above approach to be carried out. The author proposes to this end a new method using the resistance forms and the Phragmen-Lindelöf principle. As a result, he proves that the self-similar structure of the resolvent on blowups, improves the known upper estimates for the resovent kernel and estimates for other spectral operators. In addition, this work gives a new understanding of functions of the Laplacian (for instance, the heat operator \(e^{t\Delta}\)) as integrals of the resolvent.