Equivalent Semi-Norms of Non-Local Dirichlet Forms on the Sierpi\'nski Gasket and Applications

We construct equivalent semi-norms of non-local Dirichlet forms on the Sierpi\'nski gasket and apply these semi-norms to a convergence problem and a trace problem. We also construct explicitly a sequence of non-local Dirichlet forms with jumping kernels equivalent to $|x-y|^{-\alpha-\beta}$ that converges exactly to local Dirichlet form.