On the sharp upper bound related to the weak Muckenhoupt–Wheeden conjecture

Lerner, Andrei K.; Nazarov, Fedor; Ombrosi, Sheldy

doi:10.2140/apde.2020.13.1939

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On the sharp upper bound related to the weak Muckenhoupt–Wheeden conjecture

Andrei K. Lerner, Fedor Nazarov and Sheldy Ombrosi

Vol. 13 (2020), No. 6, 1939–1954

DOI: 10.2140/apde.2020.13.1939

Abstract

We construct an example showing that the upper bound ${[w]}_{A_{1}} log (e + {[w]}_{A_{1}})$ for the $L^{1} (w) \to L^{1, \infty} (w)$ norm of the Hilbert transform cannot be improved in general.

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Keywords

Hilbert transform, maximal operator, weighted inequalities

Mathematical Subject Classification 2010

Primary: 42B20, 42B25

Milestones

Received: 21 May 2019

Accepted: 29 June 2019

Published: 12 September 2020

Authors

Andrei K. Lerner
Department of Mathematics Bar-Ilan University Ramat Gan Israel

Fedor Nazarov
Department of Mathematics Kent State University Kent, OH United States

Sheldy Ombrosi
Departamento de Matemática Universidad Nacional del Sur Bahía Blanca Argentina

Vol. 13, No. 6, 2020