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

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.