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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

A singular integral operator with rough kernel

Author(s): Dashan Fan; Yibiao Pan
Journal: Proc. Amer. Math. Soc. 125 (1997), 3695-3703.
MSC (1991): Primary 42B20
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 $1<p<\infty$.

References:

[CZ1]
A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta. Math. 88 (1952), 85-139. MR 14:637f

[CZ2]
A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289-309. MR 18:894a

[Ch]
L. Chen, On a singular integral, Studia Math., TLXXXV (1987), 61-72. MR 88i:42026

[Co]
L. Colzani, Hardy Spaces on Sphere, Ph.D. Thesis, Washington University, St. Louis, MO, 1982.

[CTW]
L. Colzani, M. Taibleson and G. Weiss, Maximal estimates for Cesàro and Riesz means on sphere, Indiana Univ. Math. J. 33 (1984), No. 6, 873-889. MR 86g:43012

[DR]
J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. MR 87f:42046

[Fa]
D. Fan, Restriction theorems related to atoms, Ill. Jour. Math., Vol. 40, No. 1 (1996), 13-20. CMP 96:11

[Fe]
R. Fefferman, A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), 266-270. MR 81e:42025

[FP1]
D. Fan and Y. Pan, Oscillatory integrals and atoms on the unit sphere, Manuscripta Math., 89 (1996), 179-192. CMP 96:07

[FP2]
D. Fan and Y. Pan, $L^2$ boundedness of a singular integral operator, submitted.

[Na]
J. Namazi, A Singular Integral, Ph.D. Thesis, Indiana University, Bloomington, 1984; Proc. Amer. Math. Soc. 96 (1986), 421-424. MR 87e:42023

[St]
E. M. Stein, Problems in Harmonic Analysis Related to Curvature and Oscillatory Integrals, Proc. Inter. Cong. Math., Berkeley (1986), 196-221. MR 89d:42028

[SW]
E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. MR 46:4102

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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: 10.1090/S0002-9939-97-04111-7
PII: S 0002-9939(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
Copyright of article: Copyright 1997, American Mathematical Society




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