• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.47-73
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• 2023

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 ∈ ∆_γ(ℝ₊) and Ω ∈ Wℱ_β(S^n-1) for some γ, β ∈ (1, ∞]. As applications, the corresponding results for the generalized parametric Marcinkiewicz integral operators related to the Littlewood-Paley g^*_λ functions and area integrals are also presented.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1099-1115
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• 2019

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 ${\mathcal{F}}_{\beta}(S^{n-1})$. 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.