Strong and weak type estimates for singular integrals with respect to measures separated by AD-regular boundaries

TL;DR

This paper establishes new weak and strong boundedness estimates for singular integrals with respect to measures separated by AD-regular boundaries, extending previous results through novel Calderón-Zygmund decompositions.

Contribution

It introduces a new strategy using Calderón-Zygmund decompositions to generalize and extend existing boundedness results for singular integrals with measures separated by AD-regular boundaries.

Findings

Proved weak and strong boundedness estimates for singular integrals.

Extended results of Chousionis and Mattila to broader settings.

Developed a new Calderón-Zygmund decomposition technique.

Abstract

We prove weak and strong boundedness estimates for singular integrals in $\R^d$ with respect to $(d-1)$-dimensional measures separated by Ahlfors-David regular boundaries, generalizing and extending results of Chousionis and Mattila. Our proof follows a different strategy based on new Calder\'on-Zygmund decompositions which can be also used to extend a result of David.