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# $L^p$-boundedness of a singular integral operator

Published:1998-12-01
Printed: Dec 1998
• Abdelnaser J. Al-Hasan
• Dashan Fan
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## Abstract

Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be an $H^1$ function on the unit sphere satisfying the mean zero property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree $m$ satisfying $Q_m(0)=0$. We prove that the singular integral operator $$T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(|y|) \Omega(\,y) |y|^{-n} f \left( x-Q_m (|y|) y' \right) \,dy$$ is bounded in $L^p (\bR^n)$ for \$1
 Keywords: singular integral, rough kernel, Hardy space
 MSC Classifications: 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)

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