Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-21T00:02:15.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted Restriction for Curves

Published online by Cambridge University Press:  20 November 2018

Joseph D. Lakey
Joseph D. Lakey*
Affiliation:
Department of Mathematics The University of Texas Austin, Texas 78712 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove weighted norm inequalities for the Fourier transform of the form

where v is a nonnegative weight function on ℝd and ψ: [— 1,1 ] —> ℝd is a nondegenerate curve. Our results generalize unweighted (i.e. v = 1) restriction theorems of M. Christ, and two-dimensional weighted restriction theorems of C. Carton-Lebrun and H. Heinig.

Keywords

42B10Fourier transformrestriction theoremsweighted norm inequalities
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 1 , 01 March 1993 , pp. 87 - 95
Copyright
Copyright © Canadian Mathematical Society 1993

References

[BH] Benedetto, J. and Heinig, H., Weighted Hardy spaces and the Laplace transform, Lecture Notes in Math, 992 Springer- Verlag, Berlin, 1983,240–277.Google Scholar
[BS] Bloom, S. and Sampson, G., Weighted restriction theorems for the Fourier transform, preprint.Google Scholar
[C-L,H]C. Carton-Lebrun and Heinig, H., Weighted extensions of restriction theorems for the Fourier transform, Fourier Analysis and Applications, (ed. J. Byrnes), Kluwer publishers, The Netherlands, 1989.Google Scholar
[Chr] Christ, M., On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. AMS 287(1985), 223–238.Google Scholar
[Dr] Drury, S. W., Degenerate curves and harmonic analysis, Math. Proc. Camb. Phil. Soc. 108(1990), 89–96.Google Scholar
[DM] Drury, S. W. and Marshall, B. P., Fourier restriction theorems for degenerate curves, Math. Proc. Camb. Phil. Soc. 101(1987), 541–553.Google Scholar
[FS] Fournier, J. J. F. and Stewart, J., Amalgams of If and F, Bull. AMS 13(1985), 1–21.Google Scholar
[JS] Jurkat, W. and Sampson, G., On rearrangement and weight inequalities for the Fourier transform, Ind. U. Math J. 33(1984), 257–270.Google Scholar
[Mu]B Muckenhoupt, Weightednorm inequalities for the Fourier transform, Trans. AMS 276(1983), 729–742.Google Scholar
[Prl] Prestini, E., Restriction theorems for the Fourier transform to some submanifolds in Rn, Proc. Symp. Pure Math. XXXV, pt. 1, AMS Providence, RI, (1979), 101–109.Google Scholar
[Pr2], On the phase problem for space curves anda restriction theorem for space curves, Ph.Thesis, D., University of Maryland, College Park, MD, (1976),.Google Scholar
[Sj] Sjölin, P., Fourier multipliers and estimates of Fourier transforms of measures carried by smooth curves in R2, Studia Math. 51(1974), 169–182.Google Scholar
[StW] Stein, E. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N. J., 1971.Google Scholar
[Zygmund, Z. A., On Fourier coefficients and transforms of functions of two variables, Studia Math 50(1974), 189–201.Google Scholar