Causal sparse domination of Beurling maximal regularity operators

Abstract

We prove boundedness of Calderön–Zygmund operators acting in Banach function spaces on domains, defined by the L1 Carleson functional and Lq (1 < q < ∞) 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.