Causal sparse domination of Beurling maximal regularity operators

We prove boundedness of Calder\'on-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.