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

TL;DR

This paper establishes sharp weighted bounds for non-tangential maximal functions of singular integrals using novel Carleson-sparse domination techniques, advancing understanding of boundary behavior in harmonic analysis.

Contribution

Introduces a new Carleson-sparse domination approach for singular integrals, providing sharp weighted estimates for non-tangential maximal functions in boundary value problems.

Findings

Proves weak $L_1$ mapping of adjoint singular integrals from half-space to boundary.

Establishes non-standard sparse domination with Carleson averages.

Achieves sharp weighted estimates for non-tangential maximal functions.

Abstract

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 From this a non-standard sparse domination of the singular integrals is established, where averages have been replaced by Carleson averages.