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

Full-text PDF

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.

**[1]**Michael Christ,*Weak type (1,1) bounds for rough operators*, Ann. of Math. (2)**128**(1988), no. 1, 19–42. MR**951506**, 10.2307/1971461**[2]**R. R. Coifman and C. Fefferman,*Weighted norm inequalities for maximal functions and singular integrals*, Studia Math.**51**(1974), 241–250. MR**0358205****[3]**D. Cruz-Uribe, J. M. Martell, and C. Pérez,*Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture*, Adv. Math.**216**(2007), no. 2, 647–676. MR**2351373**, 10.1016/j.aim.2007.05.022**[4]**D. Cruz-Uribe and C. Pérez,*Two weight extrapolation via the maximal operator*, J. Funct. Anal.**174**(2000), no. 1, 1–17. MR**1761362**, 10.1006/jfan.2000.3570**[5]**C. Fefferman and E. M. Stein,*Some maximal inequalities*, Amer. J. Math.**93**(1971), 107–115. MR**0284802****[6]**Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden,*Weighted norm inequalities for the conjugate function and Hilbert transform*, Trans. Amer. Math. Soc.**176**(1973), 227–251. MR**0312139**, 10.1090/S0002-9947-1973-0312139-8**[7]**A. K. Lerner and S. Ombrosi,*An extrapolation theorem with applications to weighted estimates for singular integrals*(2010), available at`http://u.math.biu.ac.il/~lernera/publications.html`.**[8]**Michael T. Lacey, Stefanie Petermichl, and Maria Carmen Reguera,*Sharp 𝐴₂ inequality for Haar shift operators*, Math. Ann.**348**(2010), no. 1, 127–141. MR**2657437**, 10.1007/s00208-009-0473-y**[9]**Michael T. Lacey, Eric T. Sawyer, Chun-Yen Shen, and Ignacio Uriarte-Tuero,*Two Weight Inequalities for Hilbert Transform, Coronas and Energy Conditions*(2011), available at`http://www.arxiv.org/abs/1108.2319`.**[10]**Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero,*Two Weight Inequalities for Discrete Positive Operators*, submitted (2009), available at`http://www.arxiv.org/abs/0911.3437`.**[11]**Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero,*A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure*, A&PDE, to appear (2008), available at`http://arxiv.org/abs/0807.0246`.**[12]**Tuomas P. Hytönen, Michael T. Lacey, Henri Martikainen, Tuomas Orponen, Maria Carmen Reguera, Eric T. Sawyer, and Ignacio Uriarte-Tuero,*Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on 𝐴_{𝑝} weighted spaces*, J. Anal. Math.**118**(2012), no. 1, 177–220. MR**2993026**, 10.1007/s11854-012-0033-3**[13]**Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero,*A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis*(2010), available at`http://www.arXiv.org/abs/1001.4043`.**[14]**Andrei K. Lerner, Sheldy Ombrosi, and Carlos Pérez,*Sharp 𝐴₁ bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden*, Int. Math. Res. Not. IMRN**6**(2008), Art. ID rnm161, 11. MR**2427454**, 10.1093/imrn/rnm161**[15]**F. Nazarov, A. Reznikov, V. Vasuynin, and A. Volberg,*Conjecture: weak norm estimates of weighted singular operators and Bellman functions*(2010), available at`http://sashavolberg.files.wordpress.com/2010/11/a11_7loghilb11_21_2010.pdf`.**[16]**F. Nazarov, S. Treil, and A. Volberg,*Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures*(2005), available at`http://arxiv.org/abs/1003.1596`.**[17]**C. Pérez,*Weighted norm inequalities for singular integral operators*, J. London Math. Soc. (2)**49**(1994), no. 2, 296–308. MR**1260114**, 10.1112/jlms/49.2.296**[18]**Maria Carmen Reguera,*On Muckenhoupt-Wheeden conjecture*, Adv. Math.**227**(2011), no. 4, 1436–1450. MR**2799801**, 10.1016/j.aim.2011.03.009**[19]**M. C. Reguera and C. Thiele,*The Hilbert transform does not map to*(2010), available at`http://arxiv.org/abs/1011.1767`.**[20]**Eric T. Sawyer,*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), no. 1, 1–11. MR**676801****[21]**Elias M. Stein,*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
42B20,
42B25,
42B35

Retrieve articles in all journals with MSC (2010): 42B20, 42B25, 42B35

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

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.