一类新型BMO空间
Some New Classes of BMO Spaces
引进了弱型有界平均震荡函数空间WBMOq,1 < q < ∞,它是类似于弱型勒贝格空间Lq,∞所对应的 BMO空间. 证明了||·||*(BMO 范数)与||·||WBMOq之间的等价特征刻画. 作为应用,对于p∈(1,∞)和1/q=1/p-α/n,交换子[b,Iα] 是从Lp到Lq,∞的有界算子,当且仅当局部可积函数b属于BMO空间,其中Iα表示分数次积分算子. 另外,还引进以及学习了弱型的中心有界平均震荡空间Wq.
We introduce the weak bounded mean oscillation spaces WBMOq, 1 < q < ∞, which are the analog of weak Lebesgue spaces Lq,∞ in the setting of BMO space. It is obtained that the equivalence between the norms||·||* (the BMO norm) and||·||WBMOq. As an application, we show that the commutator[b,Iα] is bounded from Lp to Lq,∞ for some p∈(1,∞) and 1/q=1/p-α/n, if and only if b∈ BMO, where Iα is a fractional integral operator. Also, we introduce and study the weak central bounded mean oscillation spaces Wq.
BMO空间 / 交换子 / 等价性 / 分数次积分算子 / 弱型勒贝格空间 {{custom_keyword}} /
BMO space / commutator / equivalence / fractional integral / weak Lebesgue space {{custom_keyword}} /
[1] Chanillo S., A note on commutators, Indiana Univ. Math. J., 1982, 31: 7-16.
[2] Chen Y. Z., Lau K. S., Some new classes of Hardy spaces, J. Funct. Anal., 1989, 84: 255-278.
[3] Coifman R., Rochber R., Weiss G., Factorization theorems for Hardy spaces in several variables, Ann. of Math., 1976, 103: 611-635.
[4] Ding Y., A characterization of BMO via commutators for some operators, Northeast. Math. J., 1997, 13: 422-432.
[5] Fu Z. W., Liu Z. G., Lu S. Z., et al., Characterization for commutators of n-dimensional fractional Hardy operators, Sci. China, Ser. A, 2007, 50: 1418-1426.
[6] García-Cuerva J., Hardy spaces and Beurling algebras, J. Lond. Math. Soc., 1989, 39: 499-513.
[7] Herz C., Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech., 1968, 18: 283-324.
[8] Hardy G. H., Littlewood J. E., Pólya G., Inequalities, 2nd edn, Cambridge University Press, Cambrige, 1952.
[9] Hernández E., Yang D. C., Interpolation of Herz type Hardy spaces, Illinois J. Math., 1998, 42: 564-581.
[10] Janson S., Mean oscillation and commutators of singular integral operators, Ark. Math., 1978, 16: 263-270.
[11] Janson S., Taibleson M., Weiss G., Elementary characterization of the Morrey-Campanato spaces, Lect. Notes in Math., 1983, 992: 101-114.
[12] John F., Nirenberg L., On functions of bounded mean oscillation, Comm. Pure Appl. Math., 1961, 2: 415- 426.
[13] Komori Y., Mizuhara T., Notes on commutators and Morrey spaces, Hakkaido Math. J., 2003, 32: 345-353.
[14] Kozono H., Yamazaki M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 1994, 19: 959-1014.
[15] Long R. L., The spaces generated by blocks, Sci. China, Ser. A, 1984, 27: 16-26.
[16] Lu S. Z., Ding Y., Yan D. Y., Singular Integral and Related Topics, World Scientific, Singapore, 2007.
[17] Lu S. Z., Yang D. C., The Littlewood-Paley function and φ-transform characterizations of a new Hardy space HK2 associated with the Herz space, Studia Math., 1992, 101: 285-298.
[18] Lu S. Z., Yang D. C., Some new Hardy spaces associated with the Herz spaces and their applications (in Chinese), J. Beijing Normal Univ., 1993, 29: 10-19.
[19] Paluszyński M., Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 1995, 44: 1-17.
[20] Shi S. G., Lu S. Z., Some characterizations of Campanato spaces via commutators on Morrey spaces, Pacific J. Math., 2013, 264: 221-234.
[21] Zhang P., Multiple weighted estimates for commutators of multilinear maximal function, Acta Math. Sin. Engl. Ser., 2015, 31: 973-994.
国家自然科学基金资助项目(11661075
/
〈 | 〉 |