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Lp-Boundedness of a Singular Integral Operator

Published online by Cambridge University Press:  20 November 2018

Abdelnaser J. Al-Hasan  and
Dashan Fan
Abdelnaser J. Al-Hasan
Affiliation:
Department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee, WI 53201 USA, email: [email protected]
Dashan Fan
Affiliation:
Department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee, WI 53201 USA, email: [email protected]
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Abstract

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Let $b(t)$ be an ${{L}^{\infty }}$ function on $\mathbf{R}$, $\Omega ({y}')$ be an ${{H}^{1}}$ function on the unit sphere satisfying the mean zero property (1) and ${{Q}_{m}}(t)$ be a real polynomial on $\mathbf{R}$ of degree $m$ satisfying ${{Q}_{m}}(0)\,=\,0$. We prove that the singular integral operator

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

is bounded in ${{L}^{p}}({{\mathbf{R}}^{n}})$ for $1<p<\infty $, and the bound is independent of the coefficients of ${{Q}_{m}}(t)$.

Keywords

42B20singular integralrough kernelHardy space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 4 , 01 December 1998 , pp. 404 - 412
Copyright
Copyright © Canadian Mathematical Society 1998

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