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Fourier restriction theorems for degenerate curves

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

Fourier restriction theorems contain estimates of the form

where σ is a measure on a smooth manifold M in ∝n. This paper is a continuation of [5], which considered this problem for certain degenerate curves in ∝n. Here estimates are obtained for all curves with degeneracies of finite order. References to previous work on this problem may be found in [5].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 541 - 553
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Brown, L. D., Johnstone, I. M. and MacGibbon, K. B.. Variation diminishing transformations: a direct approach to total positi vity and its statistical applications. J. Amer. Stat. Assoc. 76 (1981), 824–832.CrossRefGoogle 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]Donoghue, W. F. Jr. Monotone Matrix Functions and Analytic Continuation (Springer-Verlag, 1974).CrossRefGoogle Scholar
[4]Drury, S. W.. Restrictions of Fourier transforms to curves. Annales de l'institut Fourier, 35 (1985), 117–123.CrossRefGoogle Scholar
[5]Drury, S. W. and Marshall, B. P.. Fourier restriction theorems for curves with affine and Euclidean arclengths. Math. Proc. Cambridge Philos. Soc. 97 (1985), 111–125.CrossRefGoogle Scholar
[6]Guggenheimer, H. W.. Differential Geometry (Dover, 1977).Google Scholar
[7]Hardy, G. H., Littlewood, J. E. and Pòlya, G.. Inequalities (Cambridge University Press, 1934).Google Scholar
[8]Hunt, R. A.. On L(p, q) spaces. L'enseignement Math. 12 (1966), 249–275.Google Scholar
[9]Karlin, S.. Total Positivity (Stanford University Press, 1968).Google Scholar
[10]Pòlya, G. and Szegö, G.. Problems and Theorems in Analysis (Springer-Verlag, 1976).CrossRefGoogle Scholar
[11]Prestini, E.. A restriction theorem for space curves. Proc. Amer. Math. Soc. 70 (1978), 8–10.CrossRefGoogle Scholar
[12]Stein, E. M.. Singular Integrals and Differentiability Properties of Functions (Princeton University Press, 1970).Google Scholar
[13]Steinig, J.. On some rules of Laguerre's and systems of equal sums of like powers. Rend. Mat. (6) 4 (1971), 629–644.Google Scholar
[14]Zygmund, A.. On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189–202.CrossRefGoogle Scholar