Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-28T19:04:29.167Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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