Convergence of Singular integrals with general measures

TL;DR

This paper proves weak convergence of L^2-bounded singular integrals with general measures in metric spaces, establishing almost everywhere convergence and principal value existence for measures with zero density.

Contribution

It extends convergence results of singular integrals to general measures and metric spaces, including principal value existence for zero density measures.

Findings

L^2-bounded singular integrals converge weakly in metric spaces

Almost everywhere convergence is established for these integrals

Principal values exist for measures with zero density

Abstract

We show that L^2-bounded singular integral in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.