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Weak and strong $ A_p$-$ A_\infty$ estimates for square functions and related operators

Authors: Tuomas P. Hytönen and Kangwei Li
Journal: Proc. Amer. Math. Soc. 146 (2018), 2497-2507
MSC (2010): Primary 42B25
Published electronically: February 28, 2018
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Abstract: We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $ [w]_{A_p}^{1/p}[w]_{A_\infty }^{1/2-1/p}\lesssim [w]_{A_p}^{1/2}$ for the weak type norm of square functions on $ L^p(w)$ for $ p>2$; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.

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Additional Information

Tuomas P. Hytönen
Affiliation: Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland

Kangwei Li
Affiliation: Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland
Address at time of publication: BCAM–Basque Center for Applied Mathematics, Mazarredo, 14. 48009 Bilbao, Basque Country, Spain
Email:, \,\,

Keywords: $A_p$-$A_\infty$ estimates, square functions
Received by editor(s): September 22, 2015
Received by editor(s) in revised form: July 26, 2017
Published electronically: February 28, 2018
Additional Notes: The authors were supported by the European Union through the ERC Starting Grant “Analytic-probabilistic methods for borderline singular integrals”. They are members of the Finnish Centre of Excellence in Analysis and Dynamics Research.
Communicated by: Ken Ono
Article copyright: © Copyright 2018 American Mathematical Society

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