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

Abstract

One of the main features of analysis on post-critically finite self-similar (pcfss) sets is that it is possible to understand the behavior of the Laplacian and its inverse, the Green operator, in terms of the self-similar structure of the set. Indeed, a major step in the approach to analysis on self-similar fractals via Dirichlet forms was Kigami’s proof [8, 10] that for a self-similar Dirichlet form the Green kernel can be written explicitly as a series in which each term is a rescaling of a single expression via the self-similar structure. In [6] this result was extended to show that the resolvent kernel of the Laplacian, meaning the kernel of (z − ∆)−1, can also be written as a self-similar series for suitable values of z ∈ C. Part of the motivation for that work was that it gives a new understanding of functions of the Laplacian (such as the heat operator et∆) by writing them as integrals of the resolvent. The purpose of the present work is to establish estimates that permit the above approach to be carried out. We study the functions occurring in the series decomposition from [6] (see Theorem 3.3 below for this decomposition) and give estimates on their decay. From this we determine estimates on the resolvent kernel and on kernels of operators defined as integrals of the resolvent kernel. In particular we recover the sharp upper estimates for the heat kernel (see Theorem 10.2) that were proved for pcfss sets by Hambly and Kumagai [4] by probabilistic methods (see also [1, 13, 3] for earlier results of this type on less general classes of sets). It is important to note that the preceding authors were able to prove not just upper estimates but also lower bounds for the heat kernel, and therefore were able to prove sharpness of their bounds. Our methods permit sharp bounds for the resolvent kernel on the positive real axis, but we do not know how to obtain these globally in the complex plane or how to obtain lower estimates for the heat kernel from them. Therefore in this direction our results are not as strong as those obtained in [4]. However in other directions we obtain more information than that known from heat kernel estimates, and we hope that our approach will complement the existing probabilistic methods. In particular we are able to obtain resolvent bounds on any ray in C other than the negative real axis (where the spectrum lies), while standard calculations from heat kernel bounds only give these estimates in a half-plane. A further consequence of our approach is that we extend (in Theorem 9.7) the decomposition from [6] to the case of blowups, which are non-compact sets with local structure equivalent to that of the underlying self-similar sets. The blowup of a pcfss set bears the same relation to the original set as the real line bears to the unit interval, see [17] for details. The structure of the paper is as follows. In Section 3 we recall some basic features of analysis on pcfss sets, as well as the main result of [6], which is the decomposition of the resolvent as a weighted sum of piecewise eigenfunctions. Section 4 then discusses