Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces
Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces
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DOI：
10.1017/s0027763000025691
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发表时间：
2006-03
影响因子：
0.8
通讯作者：
Yong Ding;Qingying Xue;K. Yabuta
中科院分区：
文献类型：
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作者：
Yong Ding;Qingying Xue;K. Yabuta
设g(F)、S(F)、g*λ(F)分别为f的Littlewood-Paley g函数、Lusin面积函数和Littlewood-Paley g*λ(F)函数。1990年，陈杰成和王斯雷证明了：对于一个BMO函数f，如果其中一个函数对ℝn中的一点是有限的，则它是有限的A.E.在ℝn上，且bmo有界性成立。最近，孙永忠将这一结果推广到Campanato空间(即Morrey空间、BMO空间和Lipschitz空间)。其中一人进一步改进了g*λ(F)的结果，并用Lipschitz核μρ(F)，μρS(F)和μλ*，ρ(F)处理了参数化的Marcinkiewicz函数。在这篇文章中，我们证明了当粗糙核满足Lp-Dini条件时，同样的结果也成立。
Abstract Let g(f), S(f), g * λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g * λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g* λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρ s(f) and μλ *,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp -Dini type condition.