Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions

TL;DR

This paper establishes the equivalence between resistance and heat kernel estimates for certain Dirichlet forms on fractals, introducing an inhomogeneous scaling function and analyzing self-similar measures.

Contribution

It constructs a spatially inhomogeneous scaling function for fractals and characterizes self-similar measures, advancing the understanding of heat kernel behavior on fractals.

Findings

Resistance estimates are equivalent to heat kernel estimates for certain Dirichlet forms.

A new inhomogeneous scaling function is constructed for self-similar fractals.

An example on rotated triangle fractals shows optimal heat kernel estimates can be independent of scaling exponents.

Abstract

In this paper, we focus on strongly local regular Dirichlet forms, especially those satisfying Morrey-type inequalities. We prove the equivalence between resistance estimates and heat kernel estimates in this case. Self-similar forms on fractals serve as a major application, where we construct a spatially inhomogeneous scaling function and characterize all the doubling self-similar measures. Further, on some special examples, the resistance conditions are reduced to some geometric conditions, on which a complete theory on self-similar Dirichlet spaces is established therein. In particular, we construct a concrete example on rotated triangle fractals, where the optimal heat kernel estimate is not related at all to the lower scaling exponent.