Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2023; 60(1): 47-73

Online first article January 26, 2023      Printed January 31, 2023

https://doi.org/10.4134/BKMS.b210850

Copyright © The Korean Mathematical Society.

On certain estimates for rough generalized parametric Marcinkiewicz integrals

Daiqing Zhang

Fujian University of Technology

Abstract

This paper is devoted to establishing certain $L^p$ bounds for the generalized parametric Marcinkiewicz integral operators associated to surfaces generated by polynomial compound mappings with rough kernels given by $h\in\Delta_\gamma(\mathbb{R}_{+})$ and $\Omega\in W\mathcal{F}_\beta({\rm S}^{n-1})$ for some $\gamma,\,\beta\in(1,\infty]$. 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 integral, polynomial compound curve, rough kernel, homogeneous Triebel-Lizorkin space

MSC numbers: 42B25, 47G10

Supported by: This work was supported by the Natural Science Foundation of University Union of Science and Technology Department of Fujian Province (No. 2019J01784).