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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:

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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Additional Information

Dmitry Ryabogin
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: ryabs@math.missouri.edu

Boris Rubin
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email: boris@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06242-6
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