Sharp estimates of martingale transforms in higher dimensions and applications to the Ahlfors-Beurling operator

2006/05/31 by Oliver Dragičević, Stefanie Petermichl, Dragicevic, Oliver +3
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.math/0606006

openalex publication_date 2006/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The main aspiration of this note is to construct several different Haar-type systems in euclidean spaces of higher dimensions and prove sharp Lp bounds for the corresponding martingale transforms. In dimension one this was a result of Burkholder. The motivation for working in this direction is the search for Lp estimates of the Ahlfors-Beurling operator.

Citations

• Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

Related

• On the norm of the Beurling-Ahlfors operator in several dimensions
• Laminates Meet Burkholder Functions
• Sharp A₂ estimates of Haar shifts via Bellman function
• The L^p-Operator Norm of a Quadratic Perturbation of the Real Part of\n the Ahlfors--Beurling Operator
• Multiplier theorems via martingale transforms