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

MathSciNet review:
3020857

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Abstract | References | Similar Articles | Additional Information

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 weights with , 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 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:
https://doi.org/10.1090/S0002-9939-2012-11474-1

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.