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

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.