Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singular integrals with rough kernels along real-analytic submanifolds in ${\mathbf{R}}^3$


Authors: Dashan Fan, Kanghui Guo and Yibiao Pan
Journal: Trans. Amer. Math. Soc. 355 (2003), 1145-1165
MSC (2000): Primary 42B20; Secondary 42B15, 42B25
Published electronically: November 5, 2002
MathSciNet review: 1938750
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: $L^p$ mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translates of a real-analytic submanifold in $\mathbf{R}^3$.


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

  • 1. Christ, M., Hilbert transforms along curves: I. Nilpotent groups, Ann. of Math. 122 (1985), 575-596. MR 87f:42039a
  • 2. Christ, M., Nagel, A., Stein, E. M. and Wainger, S., Singular and Maximal Radon Transforms: Analysis and Geometry, Ann. of Math. 150 (1999), 489-577. MR 2000j:42023
  • 3. Coifman, R. and Weiss, G., Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 56:6264
  • 4. Colzani, L., Hardy spaces on spheres, Ph.D. Thesis, Washington University, St. Louis, 1982.
  • 5. Connett, W. C., Singular integrals near $L^1$, Proc. Symposia Pure Math., 35 (1979), (S. Wainger and G. Weiss eds.), American Mathematical Society, Providence, RI, pp. 163-165. MR 80i:42014
  • 6. 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
  • 7. Fan, D., Restriction theorems related to atoms, Illinois Jour. Math. 40 (1996), 13-20. MR 97j:42004
  • 8. Fefferman, R., A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), 266-270. MR 81e:42025
  • 9. Fan, D. and Pan, Y., Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math. 119 (1997), 799-839. MR 99c:42029
  • 10. Fan, D., Guo, K. and Pan, Y., Singular integrals with rough kernels along real-analytic submanifolds in $\mathbf{R}^n$, Integral Equations Oper. Theory 33 (1999), 8-19. MR 99k:42031
  • 11. R. Hardt, Slicing and intersection theory for chains associated with real analytic varieties, Acta Math. 129 (1972), 75-136. MR 47:4110
  • 12. Hörmander, L., The analysis of linear partial differential operators, I, Springer-Verlag, Berlin, 1983. MR 85g:35002a
  • 13. Müller, D., Singular kernels supported on homogeneous submanifolds, J. Reine Angew. Math. 356 (1985), 90-118. MR 86i:43014
  • 14. Namazi, J., A singular integral, Proc. Amer. Math. Soc. 96 (1986), 421-424. MR 87e:42023
  • 15. Pan, Y., Boundedness of oscillatory singular integrals on Hardy spaces: II, Indiana Univ. Math. J. 41 (1992), 279-293. MR 97k:42032
  • 16. Phong, D. H. and Stein, E. M., Singular integrals related to the Radon transform and boundary value problems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), 7697-7701. MR 86j:42024
  • 17. Phong, D. H. and Stein, E. M., Hilbert integrals, singular integrals, and Radon transforms I., Acta Math. 157 (1986), 99-157. MR 88i:42028a
  • 18. Ricci, F. and Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, Jour. Funct. Anal. 73 (1987), 179-194. MR 88g:42023
  • 19. Ricci, F. and Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals II: Singular kernels supported on submanifolds, Jour. Funct. Anal. 78 (1988), 56-84. MR 89g:42030
  • 20. Ricci, F. and Weiss, G., A characterization of $H^1(\Sigma_{n-1})$, Proc. Symposia Pure Math., 35 (1979), (S. Wainger and G. Weiss eds.), American Mathematical Society, Providence, RI, pp. 289-294. MR 80m:30043
  • 21. Stein, E. M., Problems in Harmonic Analysis Related to Curvature and Oscillatory Integrals, Proc. Internat. Cong. Math., Berkeley (1986), 196-221. MR 89d:42028
  • 22. Stein, E. M., Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307-355. MR 88g:42022
  • 23. Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002
  • 24. Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239-1295. MR 80k:42023

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B20, 42B15, 42B25

Retrieve articles in all journals with MSC (2000): 42B20, 42B15, 42B25


Additional Information

Dashan Fan
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: fan@csd4.csd.uwm.edu

Kanghui Guo
Affiliation: Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804
Email: kag026f@smsu.edu

Yibiao Pan
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: yibiao+@pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03175-6
PII: S 0002-9947(02)03175-6
Keywords: Singular integrals, oscillatory integrals, Fourier transform, maximal functions, $L^p$ boundedness, rough kernels, real-analytic submanifolds
Received by editor(s): March 16, 1998
Received by editor(s) in revised form: July 14, 2002
Published electronically: November 5, 2002
Additional Notes: This work was done during the second author’s visit at the Department of Mathematics, University of Pittsburgh
The third author was partially supported by NSF Grant DMS-9622979
Article copyright: © Copyright 2002 American Mathematical Society