We prove boundedness of Calderón--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. This is joint work with Tuomas Hytönen.