Maximal operators of commutators of Bochner-Riesz means with Lipschitz functions

Abstract

This paper presents a weighted L² estimate with power weights for the maximal operator of commutators generated by compactly supported multipliers and Lipschitz functions. As an application, we study the almost convergence of the commutators, which is generated by the Bochner-Riesz means under the critical index and Lipschitz functions, for functions in L^p (p≥2).

Keywords

Maximal operator / commutator / Bochner-Riesz means / Lipschitz function

References

[1] Alvarez J, Bagby R J, Kurtz D S and Pérez C. Weighted estimates for commutators of linear operators. Studia Math, 1993, 104: 195-209

[2] Carbery A, Rubio de Francia J L, Vega L. Almost everywhere summability of Fourier integrals. J London Math Soc, 1988, 38: 513-524

[3] Chanillo S. A note on commutators. Indiana Univ Math J, 1982, 31: 7-16

[4] Cordoba A. The Kakeya maximal function and the spherical summation multiplier. Amer J Math, 1977, 99: 1-22

[5] Hoffman S. Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana Univ Math J, 1993, 42: 1-14

[6] Hu G, Lu S. The commutator of the Bochner-Riesz operator. Tôhoku Math J, 1996, 48: 259-266

[7] Hu G, Lu S. The maximal operator associated with the commutator of the Bochner-Riesz operator. Beijing Math, 1996, 2: 96-106

[8] Hu G, Lu S. A weighted L² estimates for the commutator of the Bochner-Riesz operator. Proc Amer Math Soc, 1997, 125: 2867-2873

[9] Liu Z. The Lipschitz estimates for the maximal commutator of the Bochner-Riesz operator. J. Beijing Normal Univ (Natural Science Edition), 2002, 38: 160-164

[10] Ma B, Hu G. Maximal operators associated with commutators of spherical means. Tohoku Math J, 1998, 50: 349-363

[11] Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ Math J, 1995, 44: 1-17

[12] Shi X, Sun Q. Weighted norm inequalities for Bochner-Riesz operators and singular integral operators. Proc Amer Math Soc, 1992, 116: 665-673

[13] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonolity and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

[14] Xia X. Boundedness of some singular integral operators and their commutators. <DissertationTip/> Thesis, Beijing Normal Univ, 2007

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