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$L^{p}$ BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  17 August 2015

FENG LIU  and
SUZHEN MAO
FENG LIU
Affiliation:
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China email liufeng860314@163.com
SUZHEN MAO*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China email suzhen.860606@163.com
*
✉suzhen.860606@163.com
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Abstract

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In an extrapolation argument, we prove certain $L^{p}\,(1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function $h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\it\lambda}}^{\ast }$-functions are given.

Keywords

nonisotropic dilationsMarcinkiewicz integralsrough kernelsextrapolation

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 3 , December 2015 , pp. 380 - 398
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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