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Certain Operators with Rough Singular Kernels

Published online by Cambridge University Press:  20 November 2018

Jiecheng Chen ,
Dashan Fan  and
Yiming Ying
Jiecheng Chen
Affiliation:
Department of Mathematics, Zhejiang University (Xixi Campus), 310028, Hangzhou, China e-mail: [email protected]
Dashan Fan
Affiliation:
Department of Mathematical Science, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA e-mail: [email protected]
Yiming Ying
Affiliation:
Department of Mathematics, Zhejiang University (Xixi Campus), 310028, Hangzhou, China e-mail: [email protected]
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Abstract

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We study the singular integral operator

$${{T}_{\Omega ,\alpha }}f\left( x \right)\,=\,\text{p}\text{.v}\text{.}\,{{\int }_{{{R}^{n}}}}\,b\left( \left| y \right| \right)\Omega \left( {{y}'} \right){{\left| y \right|}^{-n-\alpha }}\,f\left( x\,-\,y \right)\,dy,$$

defined on all test functions $f$, where $b$ is a bounded function, $\alpha \ge 0,\,\Omega \left( {{y}'} \right)$ is an integrable function on the unit sphere ${{S}^{n-1}}$ satisfying certain cancellation conditions. We prove that, for $1\,<\,p\,<\infty$, ${{T}_{\Omega ,\alpha }}$ extends to a bounded operator from the Sobolev space $L_{\alpha }^{p}$ to the Lebesgue space ${{L}^{p}}$ with $\Omega$ being a distribution in the Hardy space ${{H}^{q}}\left( {{S}^{n-1}} \right)$ where $q=\frac{n-1}{n-1+\alpha }$. The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for ${{T}_{\Omega ,\alpha }}$ on the Hardy spaces, as well as the boundedness for the truncated maximal operator $T_{\Omega ,m}^{*}$.

Keywords

42B2042B2542B15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 3 , 01 June 2003 , pp. 504 - 532
Copyright
Copyright © Canadian Mathematical Society 2003

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