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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On joint estimates for maximal functions and singular integrals on weighted spaces


Authors: Maria Carmen Reguera and James Scurry
Journal: Proc. Amer. Math. Soc. 141 (2013), 1705-1717
MSC (2010): Primary 42B20; Secondary 42B25, 42B35
Published electronically: November 19, 2012
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Abstract: We consider a conjecture attributed to Muckenhoupt and
Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for $ A_p$ weights with $ 1 < p < \infty $, through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding $ L^p$ spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of nonzero measure and exploit this lack of strict positivity in our constructions. These types of weights and techniques have been explored previously by the first author and independently with C. Thiele.


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Additional Information

Maria Carmen Reguera
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Address at time of publication: Centre for Mathematical Sciences, University of Lund, Lund, Sweden
Email: mreguera@math.gatech.edu, mreguera@maths.lth.se

James Scurry
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: jscurry3@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11474-1
PII: S 0002-9939(2012)11474-1
Keywords: Weights, Calderón-Zygmund operators
Received by editor(s): September 9, 2011
Published electronically: November 19, 2012
Additional Notes: The first author’s research was supported in part by grant NSF-DMS 0968499
The second author’s research was supported in part by the National Science Foundation under grant No. 1001098
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.