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), 17051717
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 HardyLittlewood 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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Carmen Reguera, Eric
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K. Lerner, Sheldy
Ombrosi, and Carlos
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CalderónZygmund operators and the relationship with a problem of
Muckenhoupt and Wheeden, Int. Math. Res. Not. IMRN 6
(2008), Art. ID rnm161, 11. MR 2427454
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F. Nazarov, S. Treil, and A. Volberg, Two weight estimate for the Hilbert transform and corona decomposition for nondoubling measures (2005), available at http://arxiv.org/abs/1003.1596.
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Pérez, Weighted norm inequalities for singular integral
operators, J. London Math. Soc. (2) 49 (1994),
no. 2, 296–308. MR 1260114
(94m:42037), http://dx.doi.org/10.1112/jlms/49.2.296
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Maria
Carmen Reguera, On MuckenhouptWheeden conjecture, Adv. Math.
227 (2011), no. 4, 1436–1450. MR 2799801
(2012d:42038), http://dx.doi.org/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 twoweight norm inequality for
maximal operators, Studia Math. 75 (1982),
no. 1, 1–11. MR 676801
(84i:42032)
 [21]
Elias
M. Stein, Harmonic analysis: realvariable 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
(95c:42002)
 [1]
 Michael Christ, Weak type bounds for rough operators, The Annals of Mathematics 128 (1988), no. 1, 1942. MR 951506 (89m:42013)
 [2]
 R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241250. MR 0358205 (50:10670)
 [3]
 D. CruzUribe, J. M. Martell, and C. Pérez, Sharp twoweight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture, Adv. Math. 216 (2007), no. 2, 647676. MR 2351373 (2008k:42029)
 [4]
 D. CruzUribe and C. Pérez, Two weight extrapolation via the maximal operator, J. Funct. Anal. 174 (2000), no. 1, 117. MR 1761362 (2001g:42040)
 [5]
 C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107115. MR 0284802 (44:2026)
 [6]
 Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227251. MR 0312139 (47:701)
 [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]
 M. T. Lacey, S. Petermichl, and M.C. Reguera, Sharp inequality for Haar shift operators, Math. Ann. 348 (2010), no. 1, 127141. MR 2657437 (2011i:42027)
 [9]
 Michael T. Lacey, Eric T. Sawyer, ChunYen Shen, and Ignacio UriarteTuero, 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 UriarteTuero, 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 UriarteTuero, 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]
 T. Hytönen, M. Lacey, H. Martikainen, T. Orponen, M. C. Reguera, E. Sawyer, and I. UriarteTuero, Weak and strong type estimates for maximal truncations of CalderónZygmund operators on weighted spaces, J. Anal. Math. 118 (2012), 177220. MR 2993026
 [13]
 Michael T. Lacey, Eric T. Sawyer, and Ignacio UriarteTuero, 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 bounds for CalderónZygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm161, 11. MR 2427454 (2009e:42030)
 [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 nondoubling 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, 296308. MR 1260114 (94m:42037)
 [18]
 M. C. Reguera, On MuckenhouptWheeden Conjecture, Advances in Mathematics 227 (2011), no. 4, 14361450. MR 2799801
 [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 twoweight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 111. MR 676801 (84i:42032)
 [21]
 Elias M. Stein, Harmonic analysis: realvariable 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 (95c:42002)
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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/S000299392012114741
PII:
S 00029939(2012)114741
Keywords:
Weights,
CalderónZygmund 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 NSFDMS 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.
