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 $\mathbf{R}^n$ to the half-space in $\mathbf{R}^{1+n}$ above $\mathbf{R}^n$. 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 $L_1$ functions in the half-space to weak $L_1$ 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 From this a non-standard sparse domination of the singular integrals is established, where averages have been replaced by Carleson averages.