Dyadic harmonic analysis and weighted inequalities: the sparse revolution

Abstract

We will introduce the basics of dyadic harmonic analysis and how it can be used to obtain weighted estimates for classical Calderón-Zygmund singular integral operators and their commutators. Harmonic analysts have used dyadic models for many years as a first step towards the understanding of more complex continuous operators. In 2000 Stefanie Petermichl discovered a representation formula for the venerable Hilbert transform as an average (over grids) of dyadic shift operators, allowing her to reduce arguments to finding estimates for these simpler dyadic models. For the next decade the technique used to get sharp weighted inequalities was the Bellman function method introduced by Nazarov, Treil, and Volberg, paired with sharp extrapolation by Dragičević et al. Other methods where introduced by Hytönen, Lerner, Cruz-Uribe, Martell, Pérez, Lacey, Reguera, Sawyer, Uriarte-Tuero, involving stopping time and median oscillation arguments, precursors of the very successful domination by positive sparse operators methodology. The culmination of this work was Tuomas Hytönen's 2012 proof of the $A_2$ conjecture based on a representation formula for any Calderón-Zygmund operator as an average of appropriate dyadic operators. Since then domination by sparse dyadic operators has taken central stage and has found applications well beyond Hytönen's $A_p$ theorem. We will survey this remarkable progression and more in these lecture notes.

Lecture notes for a mini course taught by the author at the CIMPA 2017 Research School - IX Escuela Santaló: Harmonic Analysis, Geometric Measure Theory and Applications, held in Buenos Aires, Argentina. 65 pages, 3 figures

References in corpus (8)

• Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures
• The Sparse T1 Theorem
• Sparse Bounds for Oscillatory and Random Singular Integrals
• Addendum to : Orthonormal bases of regular wavelets in spaces of homogeneous type
• Bellman function technique in Harmonic Analysis. Lectures of INRIA Summer School in Antibes, June 2011
• Quantitative weighted estimates for Rubio de Francia's Littlewood--Paley square function
• Two weight commutators on spaces of homogeneous type and applications
• Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Cited by in corpus (8)

• Multi-scale sparse domination
• Quadratic sparse domination and Weighted Estimates for non-integral Square Functions
• A sparse quadratic $T1$ theorem
• The structure of weight and function classes with coprime bases
• A Structure Theorem on Intersections of General Doubling Measures and Its Applications
• Variational characterizations of weighted Hardy spaces and weighted $BMO$ spaces
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting
• Bump conditions and two-weight inequalities for commutators of fractional integrals