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The Two Weight Inequality for the Hilbert Transform: A Primer

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Book cover Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 5))

Abstract

Given a pair of weights w, σ, the two weight inequality for the Hilbert transform is of the form \(\Vert H(\sigma f)\Vert _{L^{2}(w)}\lesssim \Vert f\Vert _{L^{2}(\sigma )}\). Recent work of Lacey-Sawyer-Shen-Uriarte-Tuero and Lacey have established a conjecture of Nazarov-Treil-Volberg, giving a real-variable characterization of which pairs of weights this inequality holds, provided the pair of weights do not share a common point mass. In this paper, the characterization is proved, collecting details from across several papers; counterexamples are detailed; and areas of application are indicated.

In memory of my father, H. Elton Lacey

Research supported in part by grant NSF-DMS 0968499, a grant from the Simons Foundation (#229596 to Michael Lacey), and the Australian Research Council through grant ARC-DP120100399. The author benefited from two research programs, first ‘Operator Related Function Theory and Time-Frequency Analysis’ at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during 2012–2013, and second ‘Interactions between Analysis and Geometry’ program at IPAM, UCLA, 2013.

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Notes

  1. 1.

    In particular, they noted that the simple A 2 condition was not sufficient for the boundedness of the Hilbert transform, and conjectured that half-Poisson A 2 conditions would be sufficient, an indication of the powerful sway held by the Muckenhoupt A 2 condition in the early years of the weighted theory.

  2. 2.

    Alternatively, under the assumption of w being doubling, check that the energy satisfies \(E(w,I)\gtrsim 1\), with the implied constant depending upon the doubling constant. Thus, the necessary energy inequality implies the pivotal condition.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael T. Lacey

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  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematical and Actuarial Sciences, Roosevelt University Chicago, Chicago, Illinois, USA

    Wilfredo Urbina

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Lacey, M.T. (2017). The Two Weight Inequality for the Hilbert Transform: A Primer. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2). Association for Women in Mathematics Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-51593-9_3

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