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A singular integral operator with rough kernel


Authors: Dashan Fan and Yibiao Pan
Journal: Proc. Amer. Math. Soc. 125 (1997), 3695-3703
MSC (1991): Primary 42B20
DOI: https://doi.org/10.1090/S0002-9939-97-04111-7
MathSciNet review: 1422868
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Abstract: Let $b(y)$ be a bounded radial function and $\Omega(y')$ an $H^1$ function on the unit sphere satisfying the mean zero property. Under certain growth conditions on $\Phi(t)$, we prove that the singular integral operator

\begin{equation*}T_{\Phi,b}f(x)=\text{p.v.}\ \int _{\mathbb R^n} f(x-\Phi(|y|)y') b(y)|y|^{-n}\Omega(y')\,dy \end{equation*}

is bounded in $L^p(\mathbb R^n)$ for $1<p<\infty$.


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Additional Information

Dashan Fan
Affiliation: Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: fan@alpha1.csd.uwm.edu

Yibiao Pan
Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: yibiao@tomato.math.pitt.edu

DOI: https://doi.org/10.1090/S0002-9939-97-04111-7
Keywords: Singular integral, rough kernel, Hardy space
Received by editor(s): October 25, 1995
Received by editor(s) in revised form: August 11, 1996
Additional Notes: The second author was supported in part by a grant from the National Science Foundation.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society

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