Sharp Weighted Non-tangential Maximal Estimates via Carleson-Sparse Domination

• Published in: The Journal of geometric analysis Vol. 34; no. 12
• Main Author: Rosén, Andreas
• Format: Journal Article
• Language: English
• Published: New York Springer US 2024; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Carleson functional; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Estimates; Fourier Analysis; Global Analysis and Analysis on Manifolds; Half spaces; Integrals; Matematik; Mathematics; Mathematics and Statistics; Non-tangential maximal functional; Sharp weighted estimates; Sparse domination; Carleson functional; 42B35; 42B20; Sharp weighted estimates; 42B37; Sparse domination; Non-tangential maximal functional
• ISSN: 1050-6926; 1559-002X; 1559-002X
• DOI: 10.1007/s12220-024-01814-3

We prove sharp weighted estimates for the non-tangential maximal function of singular integrals mapping functions from
R
n
to the half-space in
R
1
+
n
above
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 From this a non-standard sparse domination of the singular integrals is established, where averages have been replaced by Carleson averages.