Geodesic metrics on fractals and applications to heat kernel estimates

It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets;we further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines a metric, and let S be the boundary of M. One of our main results is that the metrics from ga satisfy the metric chain condition if and only if a ∈ S.To determine M and S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.

Lipschitz Invariance of Critical Exponents on Besov Spaces

In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension,which plays an important role in the analysis on fractals.As an application,we show examples having different critical exponents are not Lipschitz equivalent.