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

Abstract

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.