Abstract
A version of the John-Nirenberg inequality suitable for the functions b ∈ BMO with b− ∈ L∞ is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.
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References
J. Bastero, M. Milman, F. J. Ruiz: Commutators for the maximal and sharp functions. Proc. Am. Math. Soc. 128 (2000), 3329–3334.
C. Bennett, R. A. DeVore, R. Sharpley: Weak-L∞ and BMO. Ann. Math. (2) 113 (1981), 601–611.
S. Bloom: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292 (1985), 103–122.
S. Chanillo: A note on commutators. Indiana Univ. Math. J. 31 (1982), 7–16.
R. R. Coifman, R. Rochberg, G. Weiss: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103 (1976), 611–635.
J. García-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116. North-Holland, Amsterdam, 1985.
K.-P. Ho: Characterizations of BMO by Ap weights and p-convexity. Hiroshima Math. J. 41 (2011), 153–165.
M. Izuki, T. Noi, Y. Sawano: The John-Nirenberg inequality in ball Banach function spaces and application to characterization of BMO. J. Inequal. Appl. 2019 (2019), Article ID 268, 11 pages.
S. Janson: Mean oscillation and commutators of singular integral operators. Ark. Math. 16 (1978), 263–270.
F. John, L. Nirenberg: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961), 415–426.
Á. D. Martínez, D. Spector: An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces. Adv. Nonlinear Anal. 10 (2021), 877–894.
B. Muckenhoupt: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165 (1972), 207–226.
B. Muckenhoupt, R. L. Wheeden: Weighted bounded mean oscillation and the Hilbert transform. Stud. Math. 54 (1976), 221–237.
A. Uchiyama: On the compactness of operators of Hankel type. Tohoku Math. J., II. Ser. 30 (1978), 163–171.
D. Wang: Notes on commutator on the variable exponent Lebesgue spaces. Czech. Math. J. 69 (2019), 1029–1037.
D. Wang, J. Zhou, Z. Teng: Some characterizations of BLO space. Math. Nachr. 291 (2018), 1908–1918.
D. Wang, J. Zhou, Z. Teng: Characterizations of BMO and Lipschitz spaces in terms of Ap,q weights and their applications. J. Aust. Math. Soc. 107 (2019), 381–391.
P. Zhang: Multiple weighted estimates for commutators of multilinear maximal function. Acta Math. Sin., Engl. Ser. 31 (2015), 973–994.
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We would like to thank the anonymous referee for his/her comments.
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This project has been supported by the National Natural Science Foundation of China (Nos. 11971237, 12071223, 12101010) and Natural Science Foundation of China of Anhui Province (No. 2108085QA19)
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Hu, M., Wang, D. The John-Nirenberg inequality for functions of bounded mean oscillation with bounded negative part. Czech Math J 72, 1121–1131 (2022). https://doi.org/10.21136/CMJ.2022.0362-21
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DOI: https://doi.org/10.21136/CMJ.2022.0362-21