Restriction for flat surfaces of revolution in 𝐑³

Carbery, A.; Kenig, C.; Ziesler, S.

doi:10.1090/S0002-9939-07-08689-3

Restriction for flat surfaces of revolution in ${\mathbf R}^3$
HTML articles powered by AMS MathViewer

by A. Carbery, C. Kenig and S. Ziesler

Proc. Amer. Math. Soc. 135 (2007), 1905-1914

DOI: https://doi.org/10.1090/S0002-9939-07-08689-3

Published electronically: January 9, 2007

PDF | Request permission

Abstract:

We investigate restriction theorems for hypersurfaces of revolution in $\mathbf {R}^3,$ with affine curvature introduced as a mitigating factor. Abi-Khuzam and Shayya recently showed that a Stein-Tomas restriction theorem can be obtained for a class of convex hypersurfaces that includes the surfaces $\Gamma (x)=(x,e^{-1/|x|^m}), m\geq 1.$ We enlarge their class of hypersurfaces and give a much simplified proof of their result.

References

F.Abi-Khuzam & B.Shayya, Fourier Restriction to Convex Surfaces in $\mathbf {R}^3,$ to appear in Publ.Mat.
Jong-Guk Bak, Restrictions of Fourier transforms to flat curves in $\mathbf R^2$, Illinois J. Math. 38 (1994), no. 2, 327–346. MR 1260846
Luca Brandolini, Alex Iosevich, and Giancarlo Travaglini, Spherical means and the restriction phenomenon, J. Fourier Anal. Appl. 7 (2001), no. 4, 359–372. MR 1836818, DOI 10.1007/BF02514502
Anthony Carbery and Sarah Ziesler, Restriction and decay for flat hypersurfaces, Publ. Mat. 46 (2002), no. 2, 405–434. MR 1934361, DOI 10.5565/PUBLMAT_{4}6202_{0}4
Allan Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), no. 4, 519–537. MR 620265, DOI 10.1512/iumj.1981.30.30043
Alex Iosevich, Fourier transform, $L^2$ restriction theorem, and scaling, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999), no. 2, 383–387 (English, with Italian summary). MR 1706572
Alex Iosevich and Guozhen Lu, Sharpness results and Knapp’s homogeneity argument, Canad. Math. Bull. 43 (2000), no. 1, 63–68. MR 1749949, DOI 10.4153/CMB-2000-009-7
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
Daniel M. Oberlin, A uniform Fourier restriction theorem for surfaces in $\Bbb R^3$, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1195–1199. MR 2045437, DOI 10.1090/S0002-9939-03-07289-7
H.Schulz, On the decay of the Fourier transform of measures on hypersurfaces, generated by radial functions, and related restriction theorems, unpublished (1990).
Per Sjölin, Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in $R^{2}$, Studia Math. 51 (1974), 169–182. MR 385437, DOI 10.4064/sm-51-2-169-182
Christopher D. Sogge, A sharp restriction theorem for degenerate curves in $\textbf {R}^2$, Amer. J. Math. 109 (1987), no. 2, 223–228. MR 882421, DOI 10.2307/2374572
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6

AMS Website Down

Proceedings of the American Mathematical Society

Restriction for flat surfaces of revolution in ${\mathbf R}^3$
HTML articles powered by AMS MathViewer

Abstract: