  Let $B^{\sigma}_{2, \infty}$ denote the Besov space defined on a compact set
$K \subset {\Bbb R}^d$ which is equipped with an $\alpha$-regular measure
$\mu$. The {\it critical exponent} $\sigma^*$ is the supremum of the $\sigma$
such that $B^{\sigma}_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is
well-known that for many standard self-similar sets $K$, $B^{\sigma^*}_{2,
\infty}$ are the domain of some local regular Dirichlet forms. In this paper,
we explore new situations that the underlying fractal sets admit inhomogeneous
resistance scalings, which yield two types of critical exponents. We will
restrict our consideration on the p.c.f. sets. We first develop a technique of
quotient networks to study the general theory of these critical exponents. We
then construct two asymmetric p.c.f. sets, and use them to illustrate the
theory and examine the function properties of the associated Besov spaces at
the critical exponents; the various Dirichlet forms on these fractals will also
be studied.
