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Singular integrals generated by zonal measures

Authors: Dmitry Ryabogin and Boris Rubin
Journal: Proc. Amer. Math. Soc. 130 (2002), 745-751
MSC (1991): Primary 42B20; Secondary 47G10
Published electronically: August 28, 2001
MathSciNet review: 1866029
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Abstract | References | Similar Articles | Additional Information


We study $L^p$-mapping properties of the rough singular integral operator $T_\nu f(x)=\int_0^\infty dr/r \int_{\Sigma_{n - 1}} f(x-r\theta)d\nu(\theta)$ depending on a finite Borel measure $\nu$ on the unit sphere $\Sigma_{n -1}$ in $\mathbb{R}^n$. It is shown that the conditions $\sup_{\vert\xi \vert=1} \int_{\Sigma_{n -1}} \log \;(1/\vert \theta \cdot \xi \vert) d\vert\nu\vert(\theta) < \infty$, $\nu(\Sigma_{n - 1})=0$ imply the $L^p$-boundedness of $T_\nu$ for all $1<p<\infty$ provided that $n>2$ and $\nu$ is zonal.

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  • 1. Calderón, A.P. and Zygmund, A., On singular integrals, Amer. J. Math., 78 (1956), 289-309. MR 18:894a
  • 2. Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527,
  • 3. José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
  • 4. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
  • 5. Grafakos, L. and Stefanov, A., Convolution Calderón-Zygmund singular integral operators with rough kernels, in Analysis of Divergence, Control and Management of Divergent processes, (W. O. Bray, C. V. Stanojevic eds.), Birkhäuser, Boston, (1999), 119-143. CMP 2000:09
  • 6. Loukas Grafakos and Atanas Stefanov, 𝐿^{𝑝} bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J. 47 (1998), no. 2, 455–469. MR 1647912,
  • 7. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, \cyr Integraly i ryady, “Nauka”, Moscow, 1981 (Russian). \cyrÈlementarnye funktsii. [Elementary functions]. MR 635931
  • 8. Ryabogin, D. and Rubin, B., Singular integrals generated by finite measures, Preprint No. 1, 1999, Hebrew University.
  • 9. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
  • 10. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 11. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • 12. Watson, D.K., Norm inequalities for rough Calderón-Zygmund operators, having no Fourer transform decay, 1994, preprint.

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Additional Information

Dmitry Ryabogin
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Boris Rubin
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Keywords: Singular integrals, $L^p$-boundedness
Received by editor(s): September 10, 2000
Published electronically: August 28, 2001
Additional Notes: This research was partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society