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

TL;DR

This paper develops equivalent semi-norms for non-local Dirichlet forms on the Sierpiński gasket, enabling analysis of convergence and trace problems, and constructs explicit sequences of forms converging to local forms.

Contribution

It introduces new equivalent semi-norms for non-local Dirichlet forms on fractals and constructs explicit sequences converging to local forms.

Findings

Established equivalent semi-norms for non-local Dirichlet forms.

Solved convergence and trace problems using these semi-norms.

Constructed explicit sequences converging to local Dirichlet forms.

Abstract

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.