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Fourier restriction theorems for curves with affine and Euclidean arclengths*

Published online by Cambridge University Press:  24 October 2008

S. W. Drury  and
B. P. Marshall
S. W. Drury
Affiliation:
Department of Mathematics, McGill University, Montreal, P.Q. H3A 2K6, Canada
B. P. Marshall
Affiliation:
Department of Mathematics, McGill University, Montreal, P.Q. H3A 2K6, Canada
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Extract

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequality

for every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 1 , January 1985 , pp. 111 - 125
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Christ, M.. On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, preprint.Google Scholar
[2]Donoghue, W. F. Jr. Monotone Matrix Functions and Analytic Continuation (Springer-Verlag, 1974).CrossRefGoogle Scholar
[3]Drury, S. W.. Restrictions of Fourier transforms to curves, to appear, Annales de l'institut Fourier, 35/1 (1985).CrossRefGoogle Scholar
[4]Fefferman, C.. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.CrossRefGoogle Scholar
[5]Guggenheimer, H. W.. Differential Geometry (Dover, 1977).Google Scholar
[6]Herstein, I. N.. Topics in Algebra (Blaisdell, 1964).Google Scholar
[7]Hunt, R. A.. On L(p, q) spaces, L'enseignement Math. 12 (1966), 249–275.Google Scholar
[8]Marshall, B.. On the restriction of Fourier transforms to curves, preprint.Google Scholar
[9]Prestini, E.. A restriction theorem for space curves. Proc. Amer. Math. Soc. 70 (1978), 8–10.CrossRefGoogle Scholar
[10]Ruiz, A.. On the restriction of Fourier transforms to curves. In Conference on Harmonic Analysis for A. Zygmund (Wadsworth, 1983), 186–212.Google Scholar
[11]Sjölin, P.. Fourier multipliers and estimates of Fourier transforms of measures carried by smooth curves in . Studia Math. 51 (1974), 169–182.CrossRefGoogle Scholar
[12]Stein, E. M. and Weiss, G.. Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, 1971).Google Scholar
[13]Tomas, P. A.. Restriction theorems for the Fourier transform. Proc. Sympos. Pure Math., vol. xxxv, part 1, 1979, 111–114.CrossRefGoogle Scholar
[14]Weyl, H.. The Classical Groups. (Princeton Univ. Press, 1946).Google Scholar
[15]Zygmund, A.. On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189–202.CrossRefGoogle Scholar