A rough Marcinkiewicz integral along smooth curves

Volume 9, Issue 6, pp 4450--4464 http://dx.doi.org/10.22436/jnsa.009.06.84

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Authors

Feng Liu - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China. Zunwei Fu - Department of Mathematics, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, Korea. Yanpeng Zheng - Department of Information and Telecommunications Engineering, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, Korea. Qing Yuan - Department of Mathematics, Linyi University, Linyi 276000, China.

Abstract

We consider the boundedness of a kind of nonlinear integral operators on \(L^p\) spaces. Including the parametric Marcinkiewicz integrals with rough kernels along compound curves \(\{\Phi(\varphi(|y|))y'; y\in \mathbb{R}^n\}\) with \(\Phi\) satisfying certain growth conditions and \(\varphi\) being differentiable function with monotonicity and some properties on the positive real line, we investigate the \(L^p\) bounds of these operators under the integral kernels given by the sphere functions \(\Omega\) in \(H^1(S^{n-1})\) or \(\Omega\) in \(L(\log^+ L)^{\frac{1}{2}}(S^{n-1})\) and the radial function \(h \in \Delta_\gamma(\mathbb{R}^+)\). As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood-Paley \(g^*_\lambda\) -functions are presented.

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Keywords

• Parametric Marcinkiewicz
• rough kernels
• Hardy spaces
• Fourier transform estimates.

MSC

•  42B20
•  42B25
•  47G10

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