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15 February 2019 On the p-norm of the discrete Hilbert transform
Rodrigo Bañuelos, Mateusz Kwaśnicki
Duke Math. J. 168(3): 471-504 (15 February 2019). DOI: 10.1215/00127094-2018-0047

Abstract

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob h-processes, we prove that its p-norm, 1<p<, is bounded above by the Lp-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.

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Rodrigo Bañuelos. Mateusz Kwaśnicki. "On the p-norm of the discrete Hilbert transform." Duke Math. J. 168 (3) 471 - 504, 15 February 2019. https://doi.org/10.1215/00127094-2018-0047

Information

Received: 3 November 2017; Revised: 25 June 2018; Published: 15 February 2019
First available in Project Euclid: 28 January 2019

zbMATH: 07040615
MathSciNet: MR3909902
Digital Object Identifier: 10.1215/00127094-2018-0047

Subjects:
Primary: 60J42
Secondary: 60J70

Keywords: Burkholder inequalities , discrete Hilbert transform , Gundy–Varopoulos representation , martingale transform

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 3 • 15 February 2019
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