On Singularity of Energy Measures for Symmetric Diffusions with Full Off-Diagonal Heat Kernel Estimates II: Some Borderline Examples

Abstract

AbstractWe present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpiński gaskets and each of which is equipped with a canonical strongly local regular symmetric Dirichlet form. We prove that any fractal K in this family satisfies the full off-diagonal heat kernel estimates with some space-time scale function \(\varPsi _{K}\) and the singularity of the associated energy measures with respect to the canonical volume measure (uniform distribution) on K, and also that the decay rate of \(r^{-2}\varPsi _{K}(r)\) to 0 as \(r\downarrow 0\) can be made arbitrarily slow by suitable choices of K. These results together support the energy measure singularity dichotomy conjecture [Ann. Probab. 48 (2020), no. 6, 2920–2951, Conjecture 2.15] stating that, if the full off-diagonal heat kernel estimates with space-time scale function \(\varPsi \) satisfying \(\lim _{r\downarrow 0}r^{-2}\varPsi (r)=0\) hold for a strongly local regular symmetric Dirichlet space with complete metric, then the associated energy measures are singular with respect to the reference measure of the Dirichlet space.KeywordsThin scale irregular Sierpiński gasketSingularity of energy measureSub-Gaussian heat kernel estimate2020 Mathematics Subject ClassificationPrimary 28A8031C2560G30Secondary 31E0535K0860J60