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Dyadic Harmonic Analysis and Weighted Inequalities: The Sparse Revolution

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New Trends in Applied Harmonic Analysis, Volume 2

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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 toward 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, and 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.

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Notes

  1. 1.

    https://terrytao.wordpress.com/2007/07/27/dyadic-models/.

  2. 2.

    The Salem Prize, founded by the widow of Raphael Salem, is awarded every year to a young mathematician judged to have done outstanding work in Salem’s field of interest, primarily the theory of Fourier series. The prize is considered highly prestigious and many Fields Medalists previously received Salem prize (Wikipedia).

  3. 3.

    Namely, for \(g\in L^1(\mu )\) it holds that \(\mu \{x\in \mathbb R: |g(x)|>\lambda \}|\le \frac{1}{\lambda }\Vert g\Vert _{L^1(\mu )}\) for all \(\lambda >0\), in other words if \(g\in L^1(\mu )\) then \(g\in L^{1,\infty }(\mu )\), where \(g\in L^{p,\infty }(\mu )\) means \(\Vert g\Vert _{L^{p,\infty }(\mu )} := \sup _{\lambda>0} \lambda \mu ^{1/p}\{x\in \mathbb R^d: |g(x)| >\lambda \}<\infty \).

  4. 4.

    http://www-sop.inria.fr/apics/ahpi/summerschool11/bellman_lectures_volberg-1.pdf.

  5. 5.

    See page 8 in José García-Cuerva’s eulogy for José Luis Rubio de Francia (1949–1988) [75].

  6. 6.

    Nowadays called “Sawyer’s testing conditions”.

  7. 7.

    An orthonormal wavelet basis of \(L^2(\mathbb R)\) is an orthonormal basis where all its elements are translations and dilations of a fixed function \(\psi \), called the wavelet. More precisely, a function \(\psi \in L^2(\mathbb R)\) is a wavelet if and only if the functions \(\psi _{j,k}(x)=2^{j/2}\psi (2^jx-k)\) for \(j,k\in \mathbb {Z}\) form an orthonormal basis of \(L^2(\mathbb R)\).

  8. 8.

    Recipient of the 2017 Abel Prize.

  9. 9.

    A measurable set E of finite measure is Borel regular if there is a Borel set B such that \(E\subset B\) and \(\mu (E)=\mu (B)\).

  10. 10.

    The topology induced by a quasi-metric is the largest topology \(\mathcal {T}\) such that for each \(x\in X\) the quasi-metric balls centered at x form a fundamental system of neighborhoods of x. Equivalently, a set \(\varOmega \) is open, \(\varOmega \in \mathcal {T}\), if for each \(x\in \varOmega \) there exists \(r>0\) such that the quasi-metric ball \(B(x,r)\subset \varOmega \). A set in X is closed if it is the complement of an open set.

  11. 11.

    Recall that any bounded linear operator on \(L^2(\mathbb R)\) that commutes with dilations and translations and anticommutes with reflexions must be a constant multiple of the Hilbert transform.

  12. 12.

    Carlos Kenig, an Argentinian mathematician, was elected President of the International Mathematical Union in July 2018 in the International Congress of Mathematicians (ICM) held in Brazil and for the first time in the Southern hemisphere.

  13. 13.

    Jill Pipher is the president-elect of the American Mathematical Society (AMS), and will begin a 2-year term in 2019.

  14. 14.

    As I am writing these notes, it has been announced that Coifman won the 2018 Schock Prize in Mathematics for his “fundamental contributions to pure and applied harmonic analysis.”

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Acknowledgements

I would like to thank Ursula Molter, Carlos Cabrelli, and all the organizers of the CIMPA 2017 Research School—IX Escuela Santaló: Harmonic Analysis, Geometric Measure Theory and Applications, held in Buenos Aires, Argentina from July 31 to August 11, 2017, for the invitation to give the course on which these lecture notes are based. It meant a lot to me to teach in the “Pabellón 1 de la Facultad de Ciencias Exactas”, having grown up hearing stories about the mythical Universidad de Buenos Aires (UBA) from my parents, Concepción Ballester and Victor Pereyra, and dear friends (such as Julián and Amanda Araoz, Manolo Bemporad, Mischa and Yanny Cotlar, Rebeca Guber, Mauricio and Gloria Milchberg, Cora Ratto and Manuel Sadosky, Cora Sadosky and Daniel Goldstein, and Cristina Zoltan, some sadly no longer with us.) who, like us, were welcomed in Venezuela in the late 60s and 70s, and to whom I would like to dedicate these lecture notes. Unfortunately, the flow is now being reversed as many Venezuelans of all walks of life are fleeing their country and many, among them mathematicians and scientists, are finding a home in other South American countries, in particular in Argentina. I would also like to thank the enthusiastic students and other attendants, as always; there is no course without an audience; you are always an inspiration for us. I thank the kind referee, who made many comments that greatly improved the presentation, and my former Ph.D. student David Weirich, who kindly provided the figures. Last but not least, I would like to thank my husband, who looked after our boys while I was traveling, and my family in Buenos Aires who lodged and fed me.

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  1. Department of Mathematics and Statistics, 1 University of New Mexico, 311 Terrace St. NE, MSC01 1115, Albuquerque, NM, 87131-0001, USA

    María Cristina Pereyra

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  1. Vanderbilt University, Nashville, TN, USA

    Akram Aldroubi

  2. Department of Mathematics (FCEyN), Universidad de Buenos Aires, Buenos Aires, Argentina

    Carlos Cabrelli

  3. Department of Mathematics (LAMA), Université Paris-Est Créteil, Creteil, Paris, France

    Stéphane Jaffard

  4. Department of Mathematics (FCEyN), Universidad de Buenos Aires, Buenos Aires, Argentina

    Ursula Molter

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Pereyra, M.C. (2019). Dyadic Harmonic Analysis and Weighted Inequalities: The Sparse Revolution. In: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U. (eds) New Trends in Applied Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-32353-0_7

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