Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bilinear restriction estimates for surfaces with curvatures of different signs


Author: Sanghyuk Lee
Journal: Trans. Amer. Math. Soc. 358 (2006), 3511-3533
MSC (2000): Primary 42B15
DOI: https://doi.org/10.1090/S0002-9947-05-03796-7
Published electronically: August 1, 2005
MathSciNet review: 2218987
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Recently, the sharp $L^2$-bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.


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

  • 1. B. Barcelo, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321-333. MR 0805965 (86k:42023)
  • 2. J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), 147-187. MR 1097257 (92g:42010)
  • 3. J. Bourgain, On the restriction and multiplier problems in $\mathbb R\sp 3$, in Geometric aspects of functional analysis-seminar 1989-90, Lecture Notes in Math. vol. 1469, Springer-Berlin, (1991), 179-191. MR 1122623 (92m:42017)
  • 4. J. Bourgain, Estimates for cone multipliers, Operator Theory: Advances and Applications 77 (1995), 41-60. MR 1353448 (96m:42022)
  • 5. A. Carbery, Restriction implies Bochner-Riesz for paraboloids, Math. Proc. Camb. Phil. Soc. 111 (1992), 525-529. MR 1151328 (93b:42024)
  • 6. A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), 519-537. MR 0620265 (84i:42030)
  • 7. S. Lee, Some sharp bounds for the cone multiplier of negative order in $\mathbb R^3$, Bull. London Math. Soc. 35 (2003), 373-390.MR 1960948 (2004c:42027)
  • 8. S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc. 131 (2003), 1433-1442. MR 1949873 (2003k:42035)
  • 9. S. Lee, Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators, Duke Math. Journ. 122 (2004), 105-235. MR 2046812
  • 10. E. Stein, Oscillatory integrals in Fourier analysis, in Beijing lectures in harmonic analysis, Ann. of Math. Study $\char93 $ 112, Princeton Univ. Press, Princeton (1986), 307-355. MR 0864375 (88g:42022)
  • 11. E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton (1993).MR 1232192 (95c:42002)
  • 12. T. Tao, Endpoint bilinear restriction theorems for the cone and some sharp null from estimates, Math. Z. 238 (2001), 215-268.MR 1865417 (2003a:42010)
  • 13. T. Tao, A Sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359-1384. MR 2033842 (2004m:47111)
  • 14. T. Tao, Recent progress on the restriction conjecture, Park City Proceedings, to appear.
  • 15. T. Tao and A. Vargas, A bilinear approach to cone multipliers I. Resriction estimates, Geom. Funct. Anal. 10 (2000), 185-215. MR 1748920 (2002e:42012)
  • 16. T. Tao and A. Vargas, A bilinear approach to cone multipliers. II. Application, Geom. Funct. Anal. 10 (2000), 216-258. MR 1748921 (2002e:42013)
  • 17. T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjecture, J. Amer. Math. Soc. 11 (1998), 967-1000. MR 1625056 (99f:42026)
  • 18. A. Vargas, Restriction theorems for a surface with negative curvature, Math. Z. 249 (2005), 97-111. MR 2106972 (2005f:42029)
  • 19. T. Wolff, Recent work connected with the Kakeya problem, Prospects in mathmatics, (Princeton. N.J., 1996), 129-162, Amer. Math. Soc. Providence, RI, 1999. MR 1660476 (2000d:42010)
  • 20. T. Wolff, Local smoothing type estimates on $L\sp p$ for large $p$, Geom. Funct. Anal. 10 (2000), 1237-1288. MR 1800068 (2001k:42030)
  • 21. T. Wolff, A sharp cone restriction estimate, Annals of Math. 153 (2001), 661-698. MR 1836285 (2002j:42019)

Similar Articles

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

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


Additional Information

Sanghyuk Lee
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea
Address at time of publication: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706-1388
Email: sanghyuk@postech.ac.kr, slee@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03796-7
Keywords: Fourier transform, restriction estimates
Received by editor(s): January 12, 2004
Received by editor(s) in revised form: June 24, 2004
Published electronically: August 1, 2005
Additional Notes: Research of the author was supported in part by The Interdisciplinary Research Program R01-1999-00005 (primary investigator: K.-T. Kim) of The Korea Science and Engineering Foundation.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society