DIRECTED POROSITY ON CONFORMAL ITERATED FUNCTION SYSTEMS AND WEAK CONVERGENCE OF SINGULAR INTEGRALS

The aim of the present paper is twofold. We study directed porosity in connection ;with conformal iterated function systems (CIFS) and with singular integrals. We prove that limit sets of finite CIFS are porous in a stronger sense than already known. Furthermore we use directed porosity to establish that truncated singular integral operators, with respect to general Radon measures p and kernels K, converge weakly in some dense subspaces of L-2(mu) when the support of mu belongs to abroad family of sets. This class contains many fractal sets like CIFS's limit sets.