Dirichlet forms and critical exponents on fractals

Qingsong Gu, Ka-Sing Lau

Published 2017-03-21Version 1

Let $B^{\sigma}_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ with an $\alpha$-regular measure $\mu$. The {\it critical exponent} $\sigma^*$ is the largest $\sigma$ such that $B^{\sigma^*}_{2, \infty}$ remains non-trivial. The exponent is determined by the geometry of $K$ and $\mu$. In the analysis of fractals, it is known that for many standard self-similar sets $K$, $B^{\sigma^*}_{2, \infty}$ is the domain of some local regular Dirichlet forms. In this paper, we study two anomalous p.c.f. fractals $K$. On the first $K$, we provide two constructions of the local regular Dirichlet forms that do not have $B^{\sigma^*}_{2, \infty}$ as domain; one satisfies the well-known energy self-similar identity, the other one does not, and is not a conventional kind. For the second $K$, we show that the associated Besov space has two critical exponents, which is different from the usual perception. In the proof, we first discretize the Besov norm in terms of the boundary of the p.c.f. set, then determine the critical exponents and construct the Dirichlet forms through some electrical network techniques.