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

  • JONG-GUK BAK (a1), DANIEL M. OBERLIN (a2) and ANDREAS SEEGER (a3)

Abstract

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.

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Copyright

Corresponding author

For correspondence; e-mail: seeger@math.wisc.edu

Footnotes

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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.

Footnotes

References

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[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), 13931401.
[2]Bak, J.-G., Oberlin, D. and Seeger, A., ‘Restriction of Fourier transforms to curves and related oscillatory integrals’, Amer. J. Math. to appear.
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[8]Drury, S. W. and Marshall, B., ‘Fourier restriction theorems for curves with affine and Euclidean arclengths’, Math. Proc. Cambridge Philos. Soc. 97 (1985), 111125.
[9]Drury, S. W. and Marshall, B., ‘Fourier restriction theorems for degenerate curves’, Math. Proc. Cambridge Philos. Soc. 101 (1987), 541553.
[10]Hunt, R., ‘On L(p,q) spaces’, Enseign. Math. 12 (1966), 249276.
[11]Janson, S., ‘On interpolation of multilinear operators’, in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Mathematics, 1302 (Springer, Berlin, 1988), pp. 290302.
[12]Sjölin, P., ‘Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in R 2’, Studia Math. 51 (1974), 169182.
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