  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.

Classical Analysis and ODEs;Analysis of PDEs;Functional Analysis