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

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 ${\mathrm{\Lambda }}_{\sigma }^{p,q}(K)$, are identitical. In particular, we provide concrete examples that $B_{\sigma }^{p,q}(K)={\mathrm{\Lambda }}_{\sigma }^{p,q}(K)$ with $\sigma >1$. Our method is purely analytical, and does not involve any heat kernel estimate.