Nagoya Mathematical Journal

Existence and boundedness of parametrized Marcinkiewicz integral with rough kernel on Campanato spaces

Yong Ding, Qingying Xue, and Kôzô Yabuta

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Let $g(f)$, $S(f)$, $g_{\lambda}^{*}(f)$ be the Littlewood-Paley $g$ function, Lusin area function, and Littlewood-Paley $g_{\lambda}^{*}$ function of $f$, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a $\mathrm{BMO}$ function $f$, one of the above functions is finite for a single point in $\mathbb{R}^{n}$, then it is finite a.e. on $\mathbb{R}^{n}$, and $\mathrm{BMO}$ boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, $\mathrm{BMO}$, and Lipschitz spaces). One of us improved his $g_{\lambda}^{*}$ result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel $\mu^{\rho}(f)$, $\mu_{S}^{\rho}(f)$ and $\mu_{\lambda}^{\ast, \rho}(f)$. In this paper, we show that the same results hold also in the case of rough kernel satisfying $L^{p}$-Dini type condition.

Article information

Nagoya Math. J., Volume 181 (2006), 103-148.

First available in Project Euclid: 14 March 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory

Littlewood-Paley functions Marcinkiewicz function area function Campanato space Morrey space Lipschitz space BMO


Ding, Yong; Xue, Qingying; Yabuta, Kôzô. Existence and boundedness of parametrized Marcinkiewicz integral with rough kernel on Campanato spaces. Nagoya Math. J. 181 (2006), 103--148.

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