Singular integrals on Sierpinski gaskets

Abstract

We construct a class of singular integral operators associated with homogeneous Calderón-Zygmund standard kernels on $d$-dimensional, $d<1$, Sierpinski gaskets $E_d$. These operators are bounded in $L^2({\mu }_d)$ and their principal values diverge ${\mu }_d$ almost everywhere, where ${\mu }_d$ is the natural (d-dimensional) measure on $E_d$.