Causal sparse domination of Beurling maximal regularity operators

Abstract

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 < ∞) 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.

One-sided $C_{p}$ estimates via $M^{\sharp}$ function

“…However, at this point, we believe it is worth mentioning some papers which have expanded the field of one-sided estimates in the last years. Kinnunen and Saari [13,14] studied parabolic Muckenhoupt conditions in connection with PDEs and more recently Hytönen and Rosén devoted their work [11] to causal sparse domination motivated by about maximal regularity estimates for elliptic PDEs, obtaining results related to one-sided weighted estimates for singular integrals.…”