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