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Degenerate curves and harmonic analysis

  • S. W. Drury (a1)

This article deals with several related questions in harmonic analysis which are well understood for non-degenerate curves in ℝn, but poorly understood in the degenerate case. These questions invariably involve a positive ‘reference’ measure on the curve. In the non-degenerate case the choice of measure is not particularly critical and it is usually taken to be the Euclidean arclength measure. Since the questions considered here are invariant under the group of affine motions (of determinant 1), the correct choice of reference measure is the affine arclength measure. We refer the reader to Guggenheimer [8] for information on affine geometry. When the curve has degeneracies, the choice of measure becomes critical and it is the affine arclength measure which yields the most powerful results. From the Euclidean point of view the affine arclength measure has correspondingly little mass near the degeneracies and thus compensates automatically for the poor behaviour there. This principle should also be valid for general submanifolds of ℝn but alas the affine geometry of submanifolds is itself not well understood in general.

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[1] C. Bennet and R. Sharpley . Interpolation of Operators (Academic Press, 1988).

[2] M. Christ . On the restriction of the Fourier transform to curves: endpoint results and the degenerate case. Trans. Amer. Math. Soc. 287 (1985), 223238.

[3] S. W. Drury . Restrictions of Fourier transforms to curves. Ann. Inst. Fourier (Grenoble), 35 (1985), 117123.

[7] A. Greenleaf . Principal curvature in harmonic analysis. Indiana Univ. Math. J. 30 (1981), 519537.

[9] D. M. Oberlin . Convolution estimates for some measures on curves. Proc. Amer. Math. Soc. 99 (1987), 5660.

[10] E. Prestini . A restriction theorem for space curves. Proc. Amer. Math. Soc. 70 (1978), 810.

[12] E. M. Stein and S. Wainger . Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc. 84 (1978), 12391295.

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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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