Bilinear restriction estimates for surfaces with curvatures of different signs

Lee, Sanghyuk

doi:10.1090/S0002-9947-05-03796-7

Bilinear restriction estimates for surfaces with curvatures of different signs
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Trans. Amer. Math. Soc. 358 (2006), 3511-3533 Request permission

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

Bartolome Barcelo Taberner, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), no. 1, 321–333. MR 805965, DOI 10.1090/S0002-9947-1985-0805965-8
J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
Jean Bourgain, On the restriction and multiplier problems in $\textbf {R}^3$, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 179–191. MR 1122623, DOI 10.1007/BFb0089225
J. Bourgain, Estimates for cone multipliers, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 41–60. MR 1353448
Anthony Carbery, Restriction implies Bochner-Riesz for paraboloids, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 525–529. MR 1151328, DOI 10.1017/S0305004100075599
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
Sanghyuk Lee, Some sharp bounds for the cone multiplier of negative order in ${\Bbb R}^3$, Bull. London Math. Soc. 35 (2003), no. 3, 373–390. MR 1960948, DOI 10.1112/S0024609303002005
Sanghyuk Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc. 131 (2003), no. 5, 1433–1442. MR 1949873, DOI 10.1090/S0002-9939-02-06781-3
Sanghyuk Lee, Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators, Duke Math. J. 122 (2004), no. 1, 205–232. MR 2046812, DOI 10.1215/S0012-7094-04-12217-1
E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355. MR 864375
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
Terence Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, Math. Z. 238 (2001), no. 2, 215–268. MR 1865417, DOI 10.1007/s002090100251
T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. MR 2033842, DOI 10.1007/s00039-003-0449-0
T. Tao, Recent progress on the restriction conjecture, Park City Proceedings, to appear.
T. Tao and A. Vargas, A bilinear approach to cone multipliers. I. Restriction estimates, Geom. Funct. Anal. 10 (2000), no. 1, 185–215. MR 1748920, DOI 10.1007/s000390050006
T. Tao and A. Vargas, A bilinear approach to cone multipliers. II. Applications, Geom. Funct. Anal. 10 (2000), no. 1, 216–258. MR 1748921, DOI 10.1007/s000390050007
Terence Tao, Ana Vargas, and Luis Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), no. 4, 967–1000. MR 1625056, DOI 10.1090/S0894-0347-98-00278-1
Ana Vargas, Restriction theorems for a surface with negative curvature, Math. Z. 249 (2005), no. 1, 97–111. MR 2106972, DOI 10.1007/s00209-004-0691-7
Thomas Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996) Amer. Math. Soc., Providence, RI, 1999, pp. 129–162. MR 1660476
T. Wolff, Local smoothing type estimates on $L^p$ for large $p$, Geom. Funct. Anal. 10 (2000), no. 5, 1237–1288. MR 1800068, DOI 10.1007/PL00001652
Thomas Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), no. 3, 661–698. MR 1836285, DOI 10.2307/2661365