On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates II: Some borderline examples

We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpi\'{n}ski 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 $\Psi_{K}$ 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 $r^{-2}\Psi_{K}(r)$ 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 $\Psi$ satisfying $\lim_{r\downarrow 0}r^{-2}\Psi(r)=0$ 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.