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Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces

Published online by Cambridge University Press:  11 January 2016

Yong Ding ,
Qingying Xue  and
Kôzô Yabuta
Yong Ding
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing, 100875, P. R. of Chinadingy@bnu.edu.cn
Qingying Xue*
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing, 100875, P. R. of Chinaxqyyan@sohu.com
Kôzô Yabuta
Affiliation:
School of Science and Technology, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japanyabuta@ksc.kwansei.ac.jp
*
School of Science and Technology Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japanqyxue@ksc.kwansei.ac.jp
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Abstract

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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), μρs(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.

Keywords

42B25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 181 , March 2006 , pp. 103 - 148
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

Footnotes

The first named author was supported by NSF of China (Grant No. 10571015) and DPFIHE of China (Grant No. 20050027025).

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