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

Abstract

AbstractOn post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces$B^{p,q}_\sigma (K)$and the Lipschitz–Besov spaces$\Lambda ^{p,q}_\sigma (K)$, are identical. In particular, we provide concrete examples that$B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$with$\sigma>1$. Our method is purely analytical, and does not involve heat kernel estimate.