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On a counterexample related to weighted weak type estimates for singular integrals


Authors: Marcela Caldarelli, Andrei K. Lerner and Sheldy Ombrosi
Journal: Proc. Amer. Math. Soc. 145 (2017), 3005-3012
MSC (2010): Primary 42B20, 42B25
DOI: https://doi.org/10.1090/proc/13496
Published electronically: January 6, 2017
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Abstract: We show that the Hilbert transform does not map $ L^1(M_{\Phi }w)$ to $ L^{1,\infty }(w)$ for every Young function $ \Phi $ growing more slowly than $ t\log \log ({\rm e}^{\rm e}+t)$. Our proof is based on a construction of M.C. Reguera and C. Thiele.


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Additional Information

Marcela Caldarelli
Affiliation: Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, 8000, Argentina
Email: marcela.caldarelli@uns.edu.ar

Andrei K. Lerner
Affiliation: Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel
Email: lernera@math.biu.ac.il

Sheldy Ombrosi
Affiliation: Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, 8000, Argentina
Email: sombrosi@uns.edu.ar

DOI: https://doi.org/10.1090/proc/13496
Keywords: Hilbert transform, weights, weak type estimates.
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: August 11, 2016
Published electronically: January 6, 2017
Additional Notes: The second author was supported by the Israel Science Foundation (grant No. 953/13).
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society