Equivalence of Besov spaces on p.c.f. self-similar sets

Abstract

On p.c.f. self-similar sets, of which the walk dimensions of heat kernels are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B_{\sigma }^{p,q}(K)$ and the Lipschitz-Besov spaces ${\Lambda }_{\sigma }^{p,q}(K)$, are identitical. In particular, we provide concrete examples that $B_{\sigma }^{p,q}(K)={\Lambda }_{\sigma }^{p,q}(K)$ with $\sigma >1$. Our method is purely analytical, and does not involve any heat kernel estimate.

29 pages, 8 figures

References in corpus (2)

• Brownian motion on the Sierpinski carpet
• Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

Cited by in corpus (1)

• Heat kernel-based p-energy norms on metric measure spaces