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

TL;DR

This paper proves boundedness and weak convergence of certain singular integral operators between measures separated by Lipschitz graphs, advancing understanding of measure interactions in harmonic analysis.

Contribution

It establishes boundedness of maximal singular integrals and weak convergence results for measures separated by Lipschitz graphs, under broad conditions.

Findings

Boundedness of maximal operators between measures separated by Lipschitz graphs.

Weak convergence of truncated operators in dense subspaces of L^2.

Applicability to a large class of kernels and measures.

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.