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Some remarks on the restriction of the Fourier transform to surfaces

Published online by Cambridge University Press:  24 October 2008

S. W. Drury  and
K. Guo
S. W. Drury
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, McGill University, Montreal, H3A 2K6, Canada
K. Guo
Affiliation:
Department of Mathematics, Southwest Missouri State University, Springfield, Missouri, 65804, U.S.A.
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Abstract

For a class of kernels, we prove the Lp estimate for the exotic Riesz potential, with which a restriction theorem of the Fourier transform to surfaces of half the ambient dimension is proved. A simpler proof of Barcelo's result is given. We also find that it is possible to combine the Hausdorff–Young theorem with the Fefferman–Zygmund method to prove some optimal results on the restriction theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 153 - 159
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Barcelo, B.. On the restriction of the Fourier transform to a conical surface. Trans. Amer. Math. Soc. 292 (1985), 321–333.Google Scholar
[2]Christ, M.. On the restriction of the Fourier transform to curves: endpoint results and the degenerate case. Trans. Amer. Math Soc. 287 (1985), 223–238.CrossRefGoogle Scholar
[3]Christ, M.. Restriction of the Fourier transform to submanifolds of low codimension. Ph.D. dissertation. University of Chicago (1982).Google Scholar
[4]Drury, S. W. and Guo, K.. Convolution estimates related to surfaces of half ambient dimension. Math. Proc. Cambridge Philos. Soc. 110 (1991). 151–159.Google Scholar
[5]Greenleaf, A.. Principal curvature and harmonic analysis. Indiana Univ.Math. J. 30 (1981), 519–537.CrossRefGoogle Scholar
[6]Guo, K.. On the p-thin problem for hypersurfaces of ℝn with zero Gaussian curvature. Preprint.Google Scholar
[7]Hunt, R. A.. On L(p, q) spaces. Enseign. Math. 12 (1966), 249–275.Google Scholar
[8]Prestini, E.. Restriction theorems for the Fourier transform to some manifolds in ℝn. Proc. Sympos. Pure. Math. 35 (1979). 101–109.Google Scholar
[9]Strichartz, R.. Lp estimates for integral transforms. Trans. Amer. Math. Soc. 136 (1969), 33–50.Google Scholar