Abstract
In this paper, it was proved that the commutator \(\mathcal{H}_{\beta ,b} \) generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L p1 (ℝn) to L p2 (ℝn) if and only if b is a CṀO(ℝn) function, where 1/p 1 − 1/p 2 = β/n, 1 < p 1 < ∞, 0 ⩽ β < n. Furthermore, the characterization of \(\mathcal{H}_{\beta ,b} \) on the homogenous Herz space \(\dot K_q^{\alpha ,p} \)(ℝn) was obtained.
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References
Hardy G. Note on a theorem of Hilbert. Math Z, 6: 314–317 (1920)
Anderson K, Muckenhoupt B. Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions. Studia Math, 72: 9–26 (1982)
Golubov B. Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces ReH 1 and BMO. Math Sb, 188: 1041–1054 (1997)
Hardy G, Littlewood J, Polya G. Inequalities. London/New York: Cambridge Univeristy Press, 1934
Sawyer E. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans Amer Math Soc, 1: 329–337 (1984)
Long S, Wang J. Commutators of Hardy operators. J Math Anal Appl, 274: 626–644 (2002)
Chirst M, Grafakos L. Best constants for two non-convolution inequalities. Proc Amer Math Soc, 123: 1687–1693 (1995)
Lu S, Yang D. The central BMO spaces and Littlewood-Paley operators. Approx Theory Appl, 11: 72–94 (1995)
Lu S, Yang D. The decomposition of the weighted Herz spaces and its applications. Sci China Ser A-Math, 38: 147–158 (1995)
Coifman R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 103: 611–635 (1976)
Komori Y. Notes on commutators of Hardy operators. Intern J Pure Appl Math, 7: 329–334 (2003)
Komori Y. Notes on commutators on Herz-type spaces. Arch Math, 81: 318–326 (2003)
Lu S, Yang D. The local version of H p ℝn spaces at the origin. Studia Math, 116: 103–131 (1995)
García-Cuerva J. Hardy spaces and Beurling algebras. J London Math Soc, 39: 499–513 (1989)
Chen Z, Lau K. Some new classes of Hardy spaces. J Funct Anal, 123: 1687–1693 (1995)
Hernández E, Yang D. Interpolation of Herz-type Hardy spaces. Illinois J Math, 42: 564–581 (1998)
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This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)
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Fu, Zw., Liu, Zg., Lu, Sz. et al. Characterization for commutators of n-dimensional fractional Hardy operators. SCI CHINA SER A 50, 1418–1426 (2007). https://doi.org/10.1007/s11425-007-0094-4
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DOI: https://doi.org/10.1007/s11425-007-0094-4