Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • 1. Calderón, A.P. and Zygmund, A., On singular integrals, Amer. J. Math., 78 (1956), 289-309. MR 18:894a
  • 2. Duoandikoetxea, J. and Rubio de Francia, J.L., Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. MR 87f:42046
  • 3. Garcia-Cuerva, J. and Rubio de Francia, J.L., Weighted norm inequalities and related topics, Notas de Matem. 116, North-Holland, Amsterdam, 1985. MR 87d:42023
  • 4. Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, series, and products, Academic Press, New York, 1980. MR 81g:33001
  • 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. Grafakos, L. and Stefanov, A., $L^p$ bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J. 47 (1998), 455-469. MR 99i:42019
  • 7. Prudnikov, A.P., Brychkov, Yu. A. and Marichev O. I., Integrals and series, Nauka, Moscow, 1981. MR 83b:00009
  • 8. Ryabogin, D. and Rubin, B., Singular integrals generated by finite measures, Preprint No. 1, 1999, Hebrew University.
  • 9. Stein, E.M., Harmonic analysis, real variable methods, orthogonality, and oscillation integrals, Princeton Univ. Press, Princeton, N.J., 1993. MR 95c:42002
  • 10. Stein, E.M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 44:7280
  • 11. Stein, E.M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 46:4102
  • 12. Watson, D.K., Norm inequalities for rough Calderón-Zygmund operators, having no Fourer transform decay, 1994, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42B20, 47G10

Retrieve articles in all journals with MSC (1991): 42B20, 47G10

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

American Mathematical Society