Title: Homogeneous kernels and self similar sets

Authors: Vasilis Chousionis and Mariusz Urbanski

Issue: Volume 64 (2015), Issue 2, 411-431


We consider singular integrals associated with homogeneous kernels on self-similar sets. Using ideas from Ergodic Theory, we prove, among other things, that in Euclidean spaces the principal values of singular integrals associated with real analytic, homogeneous kernels fail to exist almost everywhere on self-similar sets satisfying some separation conditions. Furthermore, in general metric groups, using similar techniques, we generalize a criterion of $L^2$-unboundedness for singular integrals on self-similar sets.