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A unified method for maximal truncated Calderón–Zygmund operators in general function spaces by sparse domination

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  22 November 2019

Theresa C. Anderson  and
Bingyang Hu
Theresa C. Anderson
Affiliation:
Department of Mathematics, Purdue University, 150 N, University St., West Lafayette, IN47907, USA (tcanderson@purdue.edu)
Bingyang Hu
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Dr., Madision, WI53705, USA (bhu32@wisc.edu)
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Abstract

In this note we give simple proofs of several results involving maximal truncated Calderón–Zygmund operators in the general setting of rearrangement-invariant quasi-Banach function spaces by sparse domination. Our techniques allow us to track the dependence of the constants in weighted norm inequalities; additionally, our results hold in ℝn as well as in many spaces of homogeneous type.

Keywords

sparse dominationweighted inequalitiesBanach function spacesmaximal truncations

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 229 - 247
Copyright
Copyright © Edinburgh Mathematical Society 2019

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References

1Alvarado, R. and Mitrea, M., Hardy spaces on Ahlfors-regular quasi metric spaces, Lecture Notes in Mathematics, Volume 2142 (Springer International Publishing, Cham, 2015).CrossRefGoogle Scholar
2Anderson, T. and Vagharshakyan, A., A simple proof of the sharp weighted estimate for Calderon–Zygmund operators on homogeneous spaces, J. Geom. Anal. 24(3) (2014), 1276–1297.CrossRefGoogle Scholar
3Bennett, C. and Sharply, R., Interpolation of operators (Academic Press, New York, 1988).Google Scholar
4Buckley, S. M., Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Am. Math. Soc. 340(1) (1993), 253–272.CrossRefGoogle Scholar
5Conde, J., A note on dyadic coverings and nondoubling Calderón–Zygmund theory, J. Math. Anal. Appl. 397(2) (2013), 785–790.CrossRefGoogle Scholar
6Conde-Alonso, J. M. and Rey, G., A pointwise estimate for positive dyadic shifts and some applications, Math. Ann. 365 (2016), 1111–1135.10.1007/s00208-015-1320-yCrossRefGoogle Scholar
7Cruz-Uribe, D., Martell, J. M. and Pérez, C., Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, Volume 215 (Birkhäuser/Springer, Basel, 2011).CrossRefGoogle Scholar
8Culiuc, A., Plinio, F. D. and Ou, Y., Uniform sparse domination of singular integrals via dyadic shifts, Math. Res. Lett. 25 (2018), 21–42.CrossRefGoogle Scholar
9Curbera, G., Cuerva, J., Martell, J. and Pérez, C., Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. in Math 203 (2006), 256–318.CrossRefGoogle Scholar
10Grafakos, L., Classical Fourier analysis, 3rd edn, Graduate Texts in Mathematics, Volume 249 (Springer, New York, 2014).Google Scholar
11Grafakos, L., Modern Fourier analysis, 3rd edn, Graduate Texts in Mathematics, Volume 250 (Springer, New York, 2014).Google Scholar
12Grafakos, L. and Kalton, N., Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), 151–180.CrossRefGoogle Scholar
13Hu, G., Yang, D. and Yang, D., Boundedness of maximal singular integral operators on spaces of homogeneous type and its applications, J. Math. Soc. Japan 59(2) (2007), 323–349.CrossRefGoogle Scholar
14Hytönen, T., The sharp weighted bound for general Calderón–Zygmund operators, Ann. Math. 175 (2012), 1473–1506.CrossRefGoogle Scholar
15Hytönen, T. and Kairema, A., Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126(1) (2012), 1–33.CrossRefGoogle Scholar
16Hytönen, T. and Pérez, C., Sharp weighted bounds involving A ∞, Anal. PDE 6(4) (2013), 777–818.CrossRefGoogle Scholar
17Hytönen, T. and Pérez, C., The L(log L)ε endpoint estimate for maximal singular integral operators, J. Math. Anal. Appl. 428 (2015), 605–626.CrossRefGoogle Scholar
18Hytönen, T., Pérez, C. and Rela, E., Sharp reverse Hölder property for A ∞ weights on spaces of homogeneous type, J. Funct. Anal. 263(12) (2012), 3883–3899.CrossRefGoogle Scholar
19Lacey, M. T., An elementary proof of the A2 bound, Israel J. Math. 217(1) (2017), 181–195.CrossRefGoogle Scholar
20Lerner, A., A simple proof of the A 2 conjecture, Int. Math. Res. Not. 14 (2013), 3159–3170.CrossRefGoogle Scholar
21Lerner, A., On an estimate of Calderón–Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161.CrossRefGoogle Scholar
22Lerner, A. K. and Nazarov, F., Intuitive dyadic calculus: the basics (arXiv:1508.05639, 2015).Google Scholar
23Maligranda, L., Orlicz spaces and interpolation, Seminars in Mathematics, Volume 5 (University Estadual de Campinas, Campinas SP, Brazil, 1989).Google Scholar
24Pérez, C., Singular integrals and weights: a modern introduction, Spring School on Function spaces and Inequalities, Paseky, June, 2013.Google Scholar
25Rao, M. and Ren, Z., Theory of Orlicz spaces (Marcel Dekker, New York, 1991).Google Scholar