Bulletin of the
Korean Mathematical Society BKMS

In the present paper, we establish certain $L^p$ bounds for the generalized parametric Marcinkiewicz integral operators associated to surfaces generated by polynomial compound mappings with rough kernels in Grafakos-Stefanov class $\mathcal{F}_\beta({\rm S}^{n-1})$. Our main results improve and generalize a result given by Al-Qassem, Cheng and Pan in 2012. As applications, the corresponding results for the generalized parametric Marcinkiewicz integral operators related to the Littlewood-Paley $g_\lambda^{*}$-functions and area integrals are also presented.

Keywords: generalized parametric Marcinkiewicz integrals, polynomial compound curves, Grafakos-Stefanov class, homogeneous Triebel-Lizorkin spaces

MSC numbers: Primary 42B20, 42B25, 42B99

Supported by: This work was supported by the NNSF of China (No. 11701333) and SP-OYSTTT-CMSS (No. Sxy2016K01)