Title: Dirichlet Forms and critical exponents on Fractals
Speaker: Ka-Sing Lau (University of Pittsburgh and The Chinese University of Hong Kong)
Abstract: For a domain $\Omega$ in ${\Bbb R}^n$ with smooth boundary, it is well-known that the Sobolev spaces $W^{1,2}(\Omega)$ and $W^{s,2}(\Omega), 0<s<1$ are function spaces that are associated with the Laplacian $\Delta$ and the fractional Laplacian $(-\Delta)^s$ respectively. In the analysis on fractals, these concepts have been extended to local and nonlocal regular Dirichlet forms on fractal sets $K$, and the associated function spaces on $K$ are Besov spaces $B^{\sigma^*}_{2, \infty}$ and $B^{\sigma}_{2, 2}, \ 0< \sigma \leq \sigma^*$. We call $\sigma^*$ the critical exponent of the family of Besov spaces. In this talk, we will discuss some of the recent developments; in particular on the Dirichlet forms on $B^{\sigma^*}_{2, \infty}$. We will consider the convergence of the Besov norms $||\cdot ||_{B^{\sigma}_{2, 2}}$ to $||\cdot ||_{B^{\sigma^*}_{2, \infty}}$, analogous to a classical theorem of Bourgain-Brezis-Mironescu on Sobolev spaces. This is a joint work of Dr. Qingsong Gu.
