A note on generalized parametric Marcinkiewicz integrals
Bull. Korean Math. Soc.
Published online August 20, 2019
Feng Liu
Shandong University of Science and Technology
Abstract : 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 greatly 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 : 42B20;42B25;42B99
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