Sharp weighted non-tangential maximal estimates via Carleson-sparse domination

We prove sharp weighted estimates for the non-tangential maximal function of singular integrals mapping functions from Rⁿ to the half-space in R¹⁺ⁿ above Rⁿ. The proof is based on pointwise sparse domination of the adjoint singular integrals that map functions from the half-space back to the boundary. It is proved that these map L₁ functions in the half-space to weak L₁ functions on the boundary. From this a non-standard sparse domination of the singular integrals is established, where averages have been replaced by Carleson averages.