Journal of the
Korean Mathematical Society JKMS

We establish a weighted norm inequality for a class of rough parametric Marcinkiewicz integral operators ${\mathcal M}_{\Omega}^{\rho }$. As an application of this inequality, we obtain weighted $L^{p}$ inequalities for a class of parametric Marcinkiewicz integral operators ${\mathcal M}_{\Omega ,\lambda }^{\ast ,\rho }$ and ${\mathcal M}_{\Omega ,S}^{\rho }$ related to the Littlewood-Paley $g_{\lambda }^{\ast }$-function and the area integral $S$, respectively.

Keywords: Marcinkiewicz integral, Littlewood-Paley $g$-function, Lusin area integral, rough kernel

MSC numbers: Primary 42B25