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Fourier restriction theorems for curves with affine and Euclidean arclengths*

  • S. W. Drury (a1) and B. P. Marshall (a1)
Abstract

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].

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References
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[1]Christ M.. On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, preprint.
[2]Donoghue W. F. Jr. Monotone Matrix Functions and Analytic Continuation (Springer-Verlag, 1974).
[3]Drury S. W.. Restrictions of Fourier transforms to curves, to appear, Annales de l'institut Fourier, 35/1 (1985).
[4]Fefferman C.. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 936.
[5]Guggenheimer H. W.. Differential Geometry (Dover, 1977).
[6]Herstein I. N.. Topics in Algebra (Blaisdell, 1964).
[7]Hunt R. A.. On L(p, q) spaces, L'enseignement Math. 12 (1966), 249275.
[8]Marshall B.. On the restriction of Fourier transforms to curves, preprint.
[9]Prestini E.. A restriction theorem for space curves. Proc. Amer. Math. Soc. 70 (1978), 810.
[10]Ruiz A.. On the restriction of Fourier transforms to curves. In Conference on Harmonic Analysis for A. Zygmund (Wadsworth, 1983), 186212.
[11]Sjölin P.. Fourier multipliers and estimates of Fourier transforms of measures carried by smooth curves in inline-graphic. Studia Math. 51 (1974), 169182.
[12]Stein E. M. and Weiss G.. Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, 1971).
[13]Tomas P. A.. Restriction theorems for the Fourier transform. Proc. Sympos. Pure Math., vol. xxxv, part 1, 1979, 111114.
[14]Weyl H.. The Classical Groups. (Princeton Univ. Press, 1946).
[15]Zygmund A.. On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189202.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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