Restrictions of Fourier transforms to curves

S. W. Drury

Annales de l'institut Fourier (1985)

  • Volume: 35, Issue: 1, page 117-123
  • ISSN: 0373-0956

Abstract

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Let x ( t ) = ( t , 1 2 t 2 , 1 6 t 3 ) a certain curve in R 3 . We investigate inequalities of the type { ∫ | f ^ ( x ( t ) ) | b d t } 1 / b ≤ C ∥ f ∥ a for f ∈ 𝒮 ( R 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.

How to cite

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Drury, S. W.. "Restrictions of Fourier transforms to curves." Annales de l'institut Fourier 35.1 (1985): 117-123. <http://eudml.org/doc/74660>.

@article{Drury1985,
abstract = {Let $x(t)=(t,\{1\over 2\} t^2,\{1\over 6\}t^3)$ a certain curve in $\{\bf R\}^ 3.$ We investigate inequalities of the type $\lbrace \int \vert \hat\{f\}(x(t))\vert ^ b dt\rbrace ^\{1/b\}\le C \Vert f\Vert _ a$ for $f\in \{\cal S\}(\{\bf R\}$ 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.},
author = {Drury, S. W.},
journal = {Annales de l'institut Fourier},
keywords = {curve; inequalities},
language = {eng},
number = {1},
pages = {117-123},
publisher = {Association des Annales de l'Institut Fourier},
title = {Restrictions of Fourier transforms to curves},
url = {http://eudml.org/doc/74660},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Drury, S. W.
TI - Restrictions of Fourier transforms to curves
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 1
SP - 117
EP - 123
AB - Let $x(t)=(t,{1\over 2} t^2,{1\over 6}t^3)$ a certain curve in ${\bf R}^ 3.$ We investigate inequalities of the type $\lbrace \int \vert \hat{f}(x(t))\vert ^ b dt\rbrace ^{1/b}\le C \Vert f\Vert _ a$ for $f\in {\cal S}({\bf R}$ 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.
LA - eng
KW - curve; inequalities
UR - http://eudml.org/doc/74660
ER -

References

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  1. [1] E.M. STEIN, Harmonic Analysis on Rn, Studies in Harmonic Analysis, 97-135, The Mathematical Association of America Studies in Mathematics vol. 13, 1976. Zbl0337.42016MR57 #990
  2. [2] P.A. THOMAS, Restriction theorems for the Fourier Transform, Proceedings of Symposia in Pure Mathematics, vol. 35 part 1 (1979), 111-114, AMS, Providence. Zbl0506.42020MR81d:42029
  3. [3] E. PRESTINI, A restriction theorem for space curves, Proc. A.M.S., 70 (1978), 8-10. Zbl0343.42014MR57 #7026
  4. [4] H.W. GUGGENHEIMER, Differential Geometry, Dover, New York, 1977. Zbl0357.53002MR58 #12737
  5. [5] A. ZYGMUND, On Fourier coefficients and transforms of functions of two variables, Studia Math., 50 (1974), 189-201. Zbl0278.42005MR52 #8788
  6. [6] A. BENEDECK and R. PANZONE, The Lp spaces with mixed norm, Duke Math J., 28 (1961), 301-324. Zbl0107.08902
  7. [7] E.M. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean Spaces, Princeton U.P., Princeton 1971. Zbl0232.42007MR46 #4102
  8. [8] R.A. HUNT, On L(p, q) spaces, L'Ens. Math., 12 (1966), 249-275. Zbl0181.40301MR36 #6921
  9. [9] G. POLYA and G. SZEGÖ, Problems and Theorems in Analysis, Springer-Verlag, New York, 1976. Zbl0338.00001

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