A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

Pérez, Carlos; Rela, Ezequiel

doi:10.1090/S0002-9939-2014-12353-7

A new quantitative two weight theorem for the Hardy-Littlewood maximal operator
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by Carlos Pérez and Ezequiel Rela

Proc. Amer. Math. Soc. 143 (2015), 641-655

DOI: https://doi.org/10.1090/S0002-9939-2014-12353-7

Published electronically: October 22, 2014

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Abstract:

A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved, improving the known ones. As a consequence, a new proof of the main results in papers by Hytönen and the first author and Hytönen, the first author and Rela is obtained which avoids the use of the sharp quantitative reverse Holder inequality for $A_{\infty }$ proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the non-empty annuli condition.

References

Hugo Aimar, Singular integrals and approximate identities on spaces of homogeneous type, Trans. Amer. Math. Soc. 292 (1985), no. 1, 135–153. MR 805957, DOI 10.1090/S0002-9947-1985-0805957-9
Hugo Aimar and Roberto A. Macías, Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type, Proc. Amer. Math. Soc. 91 (1984), no. 2, 213–216. MR 740173, DOI 10.1090/S0002-9939-1984-0740173-5
Oleksandra Beznosova and Alexander Reznikov, Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic ${L}\log {L}$ and ${A}_\infty$ constants. Preprint, arXiv:1201.0520 (2012).
Javier Duoandikoetxea, Francisco Martín-Reyes, and Sheldy Ombrosi, On the $A_\infty$ conditions for general bases. 2013. Private communication.
Nobuhiko Fujii, Weighted bounded mean oscillation and singular integrals, Math. Japon. 22 (1977/78), no. 5, 529–534. MR 481968
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Ioseb Genebashvili, Amiran Gogatishvili, Vakhtang Kokilashvili, and Miroslav Krbec, Weight theory for integral transforms on spaces of homogeneous type, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 92, Longman, Harlow, 1998. MR 1791462
Tuomas Hytönen and Carlos Pérez, Sharp weighted bounds involving $A_\infty$, Anal. PDE 6 (2013), no. 4, 777–818. MR 3092729, DOI 10.2140/apde.2013.6.777
Tuomas Hytönen, Carlos Pérez, and Ezequiel Rela, Sharp reverse Hölder property for $A_\infty$ weights on spaces of homogeneous type, J. Funct. Anal. 263 (2012), no. 12, 3883–3899. MR 2990061, DOI 10.1016/j.jfa.2012.09.013
Sergei V. Hruščev, A description of weights satisfying the $A_{\infty }$ condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (1984), no. 2, 253–257. MR 727244, DOI 10.1090/S0002-9939-1984-0727244-4
Anna Kairema, Two-weight norm inequalities for potential type and maximal operators in a metric space, Publ. Mat. 57 (2013), no. 1, 3–56. MR 3058926, DOI 10.5565/PUBLMAT_{5}7113_{0}1
Liguang Liu and Teresa Luque. A ${B}_p$ condition for the strong maximal function. Trans. Amer. Math. Soc. (to appear).
Andrei K. Lerner and Kabe Moen. Mixed $A_p$-$A_\infty$ estimates with one supremum. Preprint, arXiv:1212.0571 (2013).
Kabe Moen, Sharp one-weight and two-weight bounds for maximal operators, Studia Math. 194 (2009), no. 2, 163–180. MR 2534183, DOI 10.4064/sm194-2-4
Mieczysław Mastyło and Carlos Pérez, The maximal operators between banach function spaces, Indiana Univ. Math. J. (To appear).
Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 546295, DOI 10.1016/0001-8708(79)90012-4
C. J. Neugebauer, Inserting $A_{p}$-weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 644–648. MR 687633, DOI 10.1090/S0002-9939-1983-0687633-2
C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted $L^p$-spaces with different weights, Proc. London Math. Soc. (3) 71 (1995), no. 1, 135–157. MR 1327936, DOI 10.1112/plms/s3-71.1.135
Gladis Pradolini and Oscar Salinas, Maximal operators on spaces of homogeneous type, Proc. Amer. Math. Soc. 132 (2004), no. 2, 435–441. MR 2022366, DOI 10.1090/S0002-9939-03-07079-5
Carlos Pérez and Richard L. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal. 181 (2001), no. 1, 146–188. MR 1818113, DOI 10.1006/jfan.2000.3711
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
E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, DOI 10.2307/2374799
J. Michael Wilson, Weighted inequalities for the dyadic square function without dyadic $A_\infty$, Duke Math. J. 55 (1987), no. 1, 19–50. MR 883661, DOI 10.1215/S0012-7094-87-05502-5