Singular integrals on Sierpinski gaskets

We construct a class of singular integral operators associated with homogeneous Calder\'{o}n-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 $E_d$

Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs

We shall consider the truncated singular integral operators T_{\mu, K}^{\epsilon}f(x)=\int_{\mathbb{R}^{n}\setminus B(x,\epsilon)}K(x-y)f(y)d\mu y and related maximal operators $T_{\mu,K}^{\ast}f(x)=\underset{\epsilon >0}{\sup}| T_{\mu,K}^{\epsilon}f(x)|$. We shall prove for a large class of kernels $K$ and measures $\mu$ and $\nu$ that if $\mu$ and $\nu$ are separated by a Lipschitz graph, then $T_{\nu,K}^{\ast}:L^p(\nu)\to L^p(\mu)$ is bounded for $1<p<\infty$. We shall also show that the truncated operators $T_{\mu, K}^{\epsilon}$ converge weakly in some dense subspaces of $L^2(\mu)$ under mild assumptions for the measures and the kernels.Comment: To appear in the Bulletin of the LM

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 $\mu$ 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. This class contains many fractal sets like CIFS's limit sets

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 &lt; 1, Sierpinski gaskets E d . These operators are bounded in L 2 (µ d ) and their principal values diverge µ d almost everywhere, where µ d is the natural (d-dimensional) measure on E d