We investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.
@article{bwmeta1.element.doi-10_1515_conop-2016-0013,
author = {Robert Rahm and Scott Spencer},
title = {Entropy bump conditions for fractional maximal and integral operators},
journal = {Concrete Operators},
volume = {3},
year = {2016},
pages = {112-121},
zbl = {1346.42020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_conop-2016-0013}
}
Robert Rahm; Scott Spencer. Entropy bump conditions for fractional maximal and integral operators. Concrete Operators, Tome 3 (2016) pp. 112-121. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_conop-2016-0013/
[1] Cruz-Uribe, David, Moen, Kabe, A fractional Muckenhoupt-Wheeden theorem and its consequences, Integral Equations Operator Theory, 76, 2013, 3, 421–446 | Zbl 1275.42029
[2] Cruz-Uribe, David, Moen, Kabe, One and two weight norm inequalities for Riesz potentials, Illinois J. Math., 57, 2013, 1, 295–323 | Zbl 1297.42022
[3] Cruz-Uribe, David, Two weight norm inequalities for fractional integral operators and commutators, 2015, http://arxiv.org/abs/1412.4157
[4] Duren, Peter L., Extension of a theorem of Carleson, Bull. Amer. Math. Soc., 75, 1969, 143–146 | Zbl 0184.30503
[5] Hytönen, Tuomas P., The A2 Theorem: Remarks and Complements, 2012, http://www.arxiv.org/abs/1212.3840
[6] Lerner, Andrei K., A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. Lond. Math. Soc., 42, 2010, 5, 843–856 [WoS] | Zbl 1203.42023
[7] Lacey, Michael T., Moen, Kabe, Pérez, Carlos, Torres, Rodolfo H., Sharp weighted bounds for fractional integral operators, J. Funct. Anal., 259, 2010, 5, 1073–1097 [WoS] | Zbl 1196.42014
[8] Lacey, Michael T., Sawyer, Eric T., Uriarte-Tuero, Ignacio, Two Weight Inequalities for Discrete Positive Operators, 2009, http://arxiv.org/abs/0911.3437
[9] Lacey, Michael T., Spencer, Scott, On Entropy Bounds for Calderón–Zygmund Operators, 2, 2015, 47–52 | Zbl 1333.42021
[10] Moen, Kabe, Sharp weighted bounds without testing or extrapolation, Arch. Math. (Basel), 99, 2012, 5, 457–466 [WoS] | Zbl 1266.42037
[11] Muckenhoupt, Benjamin, Wheeden, Richard, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192, 1974, 249–258 | Zbl 0226.44007
[12] Neugebauer, C. J., Inserting Ap-weights, Proc. Amer. Math. Soc., 87, 1983, 4, 644–648 | Zbl 0521.42019
[13] Pérez, Carlos, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J., 43, 1994, 2, 31–44
[14] Pérez, Carlos, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp-spaces with different weights, Proc. London Math. Soc. (3), 71, 1995, 1, 135–157 | Zbl 0829.42019
[15] Rochberg, Richard, NWO sequences, weighted potential operators, and Schrödinger eigenvalues, Duke Math. J., 72, 1993, 1, 187–215
[16] Sawyer, Eric T., A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc., 308, 1988, 2, 533–545 | Zbl 0665.42023
[17] Sawyer, Eric T., A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75, 1982, 1, 1–11 | Zbl 0508.42023
[18] Treil, Sergei, Volberg, Alexander, Entropy conditions in two weight inequalities for singular integral operators, Adv. Math., 301, 2016, 499–548 | Zbl 06620627