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The aim of the present paper is twofold. We study directed porosity inconnection with conformal iterated function systems (CIFS) and with singularintegrals. We prove that limit sets of finite CIFS are porous in a strongersense than already known. Furthermore we use directed porosity to establishthat truncated singular integral operators, with respect to general Radonmeasures $\mu$ and kernels $K$, converge weakly in some dense subspaces of$L^2(\mu)$ when the support of $\mu$ belongs to a broad family of sets. Thisclass contains many fractal sets like CIFS's limit sets.

Directed porosity on conformal iterated function systems and weak convergence of singular integrals