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RESTRICTION OF FOURIER TRANSFORMS TO CURVES II: SOME CLASSES WITH VANISHING TORSION

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  01 August 2008

JONG-GUK BAK ,
DANIEL M. OBERLIN  and
ANDREAS SEEGER
JONG-GUK BAK
Affiliation:
Department of Mathematics and the Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang 790-784, Korea (email: bak@postech.ac.kr)
DANIEL M. OBERLIN
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA (email: oberlin@math.fsu.edu)
ANDREAS SEEGER*
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA (email: seeger@math.wisc.edu)
*
✉For correspondence; e-mail: seeger@math.wisc.edu
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Abstract

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We consider the Fourier restriction operators associated to certain degenerate curves in ℝd for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.

Keywords

restriction of Fourier transformsFourier extension operatoraffine arclength measure

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 1 - 28
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

J.B. was supported in part by grant R01-2004-000-10055-0 of the Korea Science and Engineering Foundation. D.O. was supported in part by NSF grant DMS-0552041. A.S. was supported in part by NSF grant DMS-0200186.

References

[1]Bak, J.-G. and Lee, S., ‘Estimates for an oscillatory integral operator related to restriction to space curves’, Proc. Amer. Math. Soc. 132 (2004), 1393–1401.CrossRefGoogle Scholar
[2]Bak, J.-G., Oberlin, D. and Seeger, A., ‘Restriction of Fourier transforms to curves and related oscillatory integrals’, Amer. J. Math. to appear.Google Scholar
[3]Bourgain, J., ‘Estimations de certaines fonctions maximales’, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985), 499–502.Google Scholar
[4]Carbery, A., Seeger, A., Wainger, S. and Wright, J., ‘Classes of singular integral operators along variable lines’, J. Geom. Anal. 9(4) (1999), 583–605.CrossRefGoogle Scholar
[5]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
[6]Drury, S. W., ‘Restriction of Fourier transforms to curves’, Ann. Inst. Fourier 35 (1985), 117–123.CrossRefGoogle Scholar
[7]Drury, S. W., ‘Degenerate curves and harmonic analysis’, Math. Proc. Cambridge Philos. Soc. 108 (1990), 89–96.CrossRefGoogle Scholar
[8]Drury, S. W. and Marshall, B., ‘Fourier restriction theorems for curves with affine and Euclidean arclengths’, Math. Proc. Cambridge Philos. Soc. 97 (1985), 111–125.CrossRefGoogle Scholar
[9]Drury, S. W. and Marshall, B., ‘Fourier restriction theorems for degenerate curves’, Math. Proc. Cambridge Philos. Soc. 101 (1987), 541–553.CrossRefGoogle Scholar
[10]Hunt, R., ‘On L(p,q) spaces’, Enseign. Math. 12 (1966), 249–276.Google Scholar
[11]Janson, S., ‘On interpolation of multilinear operators’, in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Mathematics, 1302 (Springer, Berlin, 1988), pp. 290–302.CrossRefGoogle Scholar
[12]Sjölin, P., ‘Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in R 2’, Studia Math. 51 (1974), 169–182.CrossRefGoogle Scholar
[13]Sogge, C. D., ‘A sharp restriction theorem for degenerate curves in R 2’, Amer. J. Math. 109(2) (1987), 223–228.CrossRefGoogle Scholar