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

Abstract

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.