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Restriction implies Bochner–Riesz for paraboloids

Published online by Cambridge University Press:  24 October 2008

Anthony Carbery
Anthony Carbery
Affiliation:
Mathematics Division, University of Sussex, Falmer, Brighton BNl 9QH
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Extract

Let Σ ⊆ ℝn be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝn (U open patches in ℝn−1). The restriction problem for Σ is the question of the a priori estimate (for f ∈ S(ℝ))

(^denoting the Fourier transform). The Bochner-Riesz problem for Σ is the question of whether the functions

define Lp-bounded Fourier multiplier operators on ℝn in the range

.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 525 - 529
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Bourgain, J.. Besicovitch type maximal functions and applications to Fourier Analysis. Geom. Funct. Anal. 1 (1991), 147–187.CrossRefGoogle Scholar
[2]Bourgain, J.. On the restriction and multiplier problem in ℝ3. In Geometric Aspects of Functional Analysis – Seminar 1989–90, Lecture Notes in Math. vol. 1469 (Springer-Verlag, 1991), pp. 179–191.Google Scholar
[3]Bourgain, J.. L p estimates for oscillatory integrals in several variables. Geom. Funct. Anal. 1 (1991), 321–374.CrossRefGoogle Scholar
[4]Carleson, L. and Sjölin, P.. Oscillatory integrals and a multiplier problem for the disc. Studia Math. 44 (1972), 287–299.CrossRefGoogle Scholar
[5]Fefferman, C.. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.CrossRefGoogle Scholar
[6]Fefferman, C.. A note on the spherical summation multipliers. Israel J. Math. 15 (1973), 44–52.CrossRefGoogle Scholar
[7]Hörmander, L.. Oscillatory integrals and multipliers on FL p. Ark. Mat. 11, (1973) 1–11.CrossRefGoogle Scholar
[8]Nikishin, E. M.. A resonance theorem and series of eigenfunctions of the Laplacian. Math. USSR Izv. 6 (1972), 788–806.CrossRefGoogle Scholar
[9]Stein, E. M.. Oscillatory integrals in Fourier analysis. In Beijing Lectures in Harmonic Analysis, Annals of Math. Studies no. 112 (Princeton University Press, 1986), pp. 307–355.Google Scholar
[10]Tomas, P., A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477–478.CrossRefGoogle Scholar
[11]Zygmund, A.. On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189–201.CrossRefGoogle Scholar