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

Reguera, Maria Carmen; Scurry, James

doi:10.1090/S0002-9939-2012-11474-1

On joint estimates for maximal functions and singular integrals on weighted spaces
HTML articles powered by AMS MathViewer

by Maria Carmen Reguera and James Scurry PDF

Proc. Amer. Math. Soc. 141 (2013), 1705-1717 Request permission

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.

References

Michael Christ, Weak type $(1,1)$ bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. MR 951506, DOI 10.2307/1971461
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
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, DOI 10.1016/j.aim.2007.05.022
D. Cruz-Uribe and C. Pérez, Two weight extrapolation via the maximal operator, J. Funct. Anal. 174 (2000), no. 1, 1–17. MR 1761362, DOI 10.1006/jfan.2000.3570
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
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 312139, DOI 10.1090/S0002-9947-1973-0312139-8
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.
Michael T. Lacey, Stefanie Petermichl, and Maria Carmen Reguera, Sharp $A_2$ inequality for Haar shift operators, Math. Ann. 348 (2010), no. 1, 127–141. MR 2657437, DOI 10.1007/s00208-009-0473-y
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.
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.
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.
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, DOI 10.1007/s11854-012-0033-3
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.
Andrei K. Lerner, Sheldy Ombrosi, and Carlos Pérez, Sharp $A_1$ 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, DOI 10.1093/imrn/rnm161
F. Nazarov, A. Reznikov, V. Vasuynin, and A. Volberg, $A_1$ 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.
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.
C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. (2) 49 (1994), no. 2, 296–308. MR 1260114, DOI 10.1112/jlms/49.2.296
Maria Carmen Reguera, On Muckenhoupt-Wheeden conjecture, Adv. Math. 227 (2011), no. 4, 1436–1450. MR 2799801, DOI 10.1016/j.aim.2011.03.009
M. C. Reguera and C. Thiele, The Hilbert transform does not map $L^{1}(Mw)$ to $L^{1,\infty }(w)$ (2010), available at http://arxiv.org/abs/1011.1767.
Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI 10.4064/sm-75-1-1-11
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