A singular integral operator with rough kernel
HTML articles powered by AMS MathViewer
- by Dashan Fan and Yibiao Pan PDF
- Proc. Amer. Math. Soc. 125 (1997), 3695-3703 Request permission
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
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Lung-Kee Chen, On a singular integral, Studia Math. 85 (1986), no. 1, 61–72 (1987). MR 879417, DOI 10.4064/sm-85-1-61-72
- L. Colzani, Hardy Spaces on Sphere, Ph.D. Thesis, Washington University, St. Louis, MO, 1982.
- Leonardo Colzani, Mitchell H. Taibleson, and Guido Weiss, Maximal estimates for Cesàro and Riesz means on spheres, Indiana Univ. Math. J. 33 (1984), no. 6, 873–889. MR 763947, DOI 10.1512/iumj.1984.33.33047
- 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, DOI 10.1007/BF01388746
- D. Fan, Restriction theorems related to atoms, Ill. Jour. Math., Vol. 40, No. 1 (1996), 13–20.
- R. Fefferman, A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), no. 2, 266–270. MR 524298, DOI 10.1090/S0002-9939-1979-0524298-3
- D. Fan and Y. Pan, Oscillatory integrals and atoms on the unit sphere, Manuscripta Math., 89 (1996), 179–192.
- D. Fan and Y. Pan, $L^2$ boundedness of a singular integral operator, submitted.
- Javad Namazi, A singular integral, Proc. Amer. Math. Soc. 96 (1986), no. 3, 421–424. MR 822432, DOI 10.1090/S0002-9939-1986-0822432-2
- E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 196–221. MR 934224
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
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
- 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 1997 American Mathematical Society
- 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