Abstract.
We prove a bilinear restriction theorem for a surface of negative curvature. This is the analogue of the results of T. Wolff [19] and T. Tao [14], [15] for cones and paraboloids. As a consequence we obtain an almost sharp linear restriction theorem.
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Barcelo, B.: On the restriction of the Fourier transform to a conical surface. Trans. Amer. Math. Soc. 292, 321–333 (1985)
Bourgain, J.: Besicovitch-type maximal operators and applications to Fourier analysis. Geom. and Funct. Anal. 22, 147–187 (1991)
Bourgain, J.: On the restriction and multiplier problem in R3. Lecture notes in Mathematics, no. 1469. Springer Verlag, 1991
Bourgain, J.: Estimates for cone multipliers, Operator Theory: Advances and Applications, 77, 41–60 (1995)
Bourgain, J.: Some new estimates on oscillatory integrals. Essays in Fourier Analysis in honor of E. M. Stein, Princeton University Press 1995, pp. 83–112
Carleson, L., Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc. Studia Math. 44, 287–299 (1972)
Feffermann, C.: Inequalities for strongly singular convolution operators. Acta Math. 44, 287–299 (1972)
Lee, S.: Bilinear Restriction Estimates for Surfaces with Curvatues of Different Signs. Preprint
Moyua, A., Vargas, A., Vega, L.: Schrödinger Maximal Function and Restriction Properties of the Fourier transform. International Math. Research Notices 16 (1996)
Moyua, A., Vargas, A., Vega, L.: Restriction theorems and Maximal operators related to oscillatory integrals in R3. To appear, Duke Math. J.
Stein, E. M.: Oscillatory integrals in Fourier analysis. Beijing Lectures in Harmonic Analysis. Annals of Math. Study #112, Princeton University Press, 1986
Strichartz, R. S.: Restriction of Fourier Transforms to quadratic surfaces and decay of solutions of wave equation. Duke Math. J. 44, 705–713 (1977)
Tao, T.: The Bochner-Riesz conjecture implies the Restriction conjecture. Duke Math. J. 96, 363–376 (1999)
Tao, T.: Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238, 215–268 (2001)
Tao, T.: A sharp bilinear restriction estimate for paraboloids. To appear, GAFA
Tao, T., Vargas, A., Vega, L.: A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc. 11, 967–1000 (1998)
Tao, T., Vargas, A.: A bilinear approach to cone multipliers I. Restriction estimates. GAFA 10, 185–215 (2000)
Tomas, P.: A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81, 477–478 (1975)
Wolff, T.: A sharp bilinear cone restriction theorem. To appear, Annals of Math.
Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 50, 189–201 (1974)
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Mathematics Subject Classification (1991):42B15
Research partially supported by EU network ‘HARP’ HPRN-CT-2001-00273, and by MCyT Grant BFM2001/0189
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Vargas, A. Restriction theorems for a surface with negative curvature. Math. Z. 249, 97–111 (2005). https://doi.org/10.1007/s00209-004-0691-7
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DOI: https://doi.org/10.1007/s00209-004-0691-7