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

Abstract

We shall consider the truncated singular integral operators T_{μ, K}^εf(x)=\int_{\mathbb{R}^{n}\setminus B(x,ε)}K(x-y)f(y)dμy 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.

To appear in the Bulletin of the LMS

References in corpus (1)

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

Cited by in corpus (1)

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