Revista Matemática Iberoamericana
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Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case

Shuanglin Shao
Source: Rev. Mat. Iberoamericana Volume 25, Number 3 (2009), 1127-1168.

Abstract

For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical \textit{linear or bilinear adjoint restriction conjectures} for such functions and verify the \textit{linear adjoint restriction conjecture} for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schrödinger equation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere.

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Primary Subjects: 42B10, 42B25
Secondary Subjects: 35Q55
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.rmi/1257258103
Zentralblatt MATH identifier: 05663372
Mathematical Reviews number (MathSciNet): MR2590695

References

Bourgain, J.: Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 1 (1991), no. 2, 147-187.
Mathematical Reviews (MathSciNet): MR1097257
Zentralblatt MATH: 0756.42014
Digital Object Identifier: doi:10.1007/BF01896376
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3 (1993), no. 2, 107-156.
Mathematical Reviews (MathSciNet): MR1209299
Digital Object Identifier: doi:10.1007/BF01896020
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation. Geom. Funct. Anal. 3 (1993), no. 3, 209-262.
Mathematical Reviews (MathSciNet): MR1215780
Zentralblatt MATH: 0787.35098
Digital Object Identifier: doi:10.1007/BF01895688
Bourgain, J.: Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications 46. American Mathematical Society, Providence, RI, 1999.
Mathematical Reviews (MathSciNet): MR1691575
Carleson, L. and Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc. Studia Math. 44 (1972), 287-299. (errata insert).
Mathematical Reviews (MathSciNet): MR361607
Cordoba, A.: The Kakeya maximal function and the spherical summation multipliers. Amer. J. Math. 99 (1977), no. 1, 1-22.
Mathematical Reviews (MathSciNet): MR447949
Zentralblatt MATH: 0384.42008
Digital Object Identifier: doi:10.2307/2374006
Fefferman, C. and Stein, E. M.: Some maximal inequalities. Amer. J. Math. 93 (1971), 107-115.
Mathematical Reviews (MathSciNet): MR284802
Digital Object Identifier: doi:10.2307/2373450
Keel, M. and Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955-980.
Mathematical Reviews (MathSciNet): MR1646048
Zentralblatt MATH: 0922.35028
Digital Object Identifier: doi:10.1353/ajm.1998.0039
Klainerman, S. and Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268.
Mathematical Reviews (MathSciNet): MR1231427
Zentralblatt MATH: 0803.35095
Digital Object Identifier: doi:10.1002/cpa.3160460902
Lee, S. and Vargas, A.: Sharp null forms estimates for the wave equation. Amer. J. Math. 130 (2008), no. 5, 1279-1326.
Mathematical Reviews (MathSciNet): MR2450209
Zentralblatt MATH: 1158.35112
Digital Object Identifier: doi:10.1353/ajm.0.0024
Moyua, A., Vargas, A. and Vega, L.: Schrödinger maximal function and restriction properties of the Fourier transform. Internat. Math. Res. Notices (1996), no. 16, 793-815.
Mathematical Reviews (MathSciNet): MR1413873
Digital Object Identifier: doi:10.1155/S1073792896000499
Moyua, A., Vargas, A. and Vega, L.: Restriction theorems and maximal operators related to oscillatory integrals in $\mathbbR\sp 3$. Duke Math. J. 96 (1999), no. 3, 547-574.
Stein, E. M.: Some problems in harmonic analysis. In Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, 3-20. Proc. Sympos. Pure Math. XXXV. Amer. Math. Soc., Providence, RI, 1979.
Mathematical Reviews (MathSciNet): MR545235
Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton, NJ, 1993.
Mathematical Reviews (MathSciNet): MR1232192
Zentralblatt MATH: 0821.42001
Stein, E. M. and Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series 32. Princeton University Press, Princeton, NJ, 1971.
Mathematical Reviews (MathSciNet): MR304972
Zentralblatt MATH: 0232.42007
Strichartz, R. S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705-714.
Mathematical Reviews (MathSciNet): MR512086
Zentralblatt MATH: 0372.35001
Digital Object Identifier: doi:10.1215/S0012-7094-77-04430-1
Project Euclid: euclid.dmj/1077312392
Tao, T.: Recent progress on the restriction conjecture. In Fourier analysis and convexity, 217-243. Appl. Numer. Harmon. Anal. Birkhäuser Boston, Boston, MA, 2004.
Mathematical Reviews (MathSciNet): MR2087245
Zentralblatt MATH: 1083.42008
Tao, T.: The Bochner-Riesz conjecture implies the restriction conjecture. Duke Math. J. 96 (1999), no. 2, 363-375.
Mathematical Reviews (MathSciNet): MR1666558
Zentralblatt MATH: 0980.42006
Digital Object Identifier: doi:10.1215/S0012-7094-99-09610-2
Project Euclid: euclid.dmj/1077229137
Tao, T.: Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238 (2001), no. 2, 215-268.
Mathematical Reviews (MathSciNet): MR1865417
Zentralblatt MATH: 0992.42004
Digital Object Identifier: doi:10.1007/s002090100251
Tao, T.: A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13 (2003), no. 6, 1359-1384.
Mathematical Reviews (MathSciNet): MR2033842
Digital Object Identifier: doi:10.1007/s00039-003-0449-0
Tao, T.: A counterexample to an endpoint bilinear Strichartz inequality. Electron. J. Differential Equations (2006), no. 151, 6 pp. (electronic)
Mathematical Reviews (MathSciNet): MR2276576
Zentralblatt MATH: 1128.35315
Tao, T. and Vargas, A.: A bilinear approach to cone multipliers. II. Applications. Geom. Funct. Anal. 10 (2000), no. 1, 216-258.
Mathematical Reviews (MathSciNet): MR1748921
Zentralblatt MATH: 0949.42013
Digital Object Identifier: doi:10.1007/s000390050007
Tao, T., Vargas, A. and Vega, L.: A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc. 11 (1998), no. 4, 967-1000.
Mathematical Reviews (MathSciNet): MR1625056
Zentralblatt MATH: 0924.42008
Digital Object Identifier: doi:10.1090/S0894-0347-98-00278-1
Tao, T., Visan, M. and Zhang, X.: Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions. Duke Math. J. 140 (2007), no. 1, 165-202.
Mathematical Reviews (MathSciNet): MR2355070
Digital Object Identifier: doi:10.1215/S0012-7094-07-14015-8
Project Euclid: euclid.dmj/1190730777
Tomas, P. A.: A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477-478.
Mathematical Reviews (MathSciNet): MR358216
Zentralblatt MATH: 0298.42011
Digital Object Identifier: doi:10.1090/S0002-9904-1975-13790-6
Project Euclid: euclid.bams/1183536437
Vilela, M. C.: Regularity of solutions to the free Schrödinger equation with radial initial data. Illinois J. Math. 45 (2001), no. 2, 361-370.
Mathematical Reviews (MathSciNet): MR1878609
Project Euclid: euclid.ijm/1258138345
Wolff, T.: An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11 (1995), no. 3, 651-674.
Mathematical Reviews (MathSciNet): MR1363209
Zentralblatt MATH: 0848.42015
Wolff, T.: A sharp bilinear cone restriction estimate. Ann. of Math. (2) 153 (2001), no. 3, 661-698.
Mathematical Reviews (MathSciNet): MR1836285
Zentralblatt MATH: 1125.42302
Digital Object Identifier: doi:10.2307/2661365
Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189-201.
Mathematical Reviews (MathSciNet): MR387950
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