#### Vol. 13, No. 6, 2020

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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
##### Abstract

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

##### 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