DIRICHLET FORMS AND CRITICAL EXPONENTS ON FRACTALS

Let B-2,infinity(sigma) denote the Besov space defined on a compact set K subset of R-d which is equipped with an alpha-regular measure it. The critical exponent sigma* is the supremum of the sigma such that B-2,infinity(sigma) boolean AND C(K) is dense in C(K). It is well known that for many standard self-similar sets K, B-2,infinity(sigma) 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. (post critically finite) 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.