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Estimates for an oscillatory integral operator related to restriction to space curves


Authors: Jong-Guk Bak and Sanghyuk Lee
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1393-1401
MSC (2000): Primary 42B10
DOI: https://doi.org/10.1090/S0002-9939-03-07144-2
Published electronically: December 5, 2003
MathSciNet review: 2053345
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the oscillatory integral operator defined by

\begin{displaymath}T_\lambda f(x)=\int_{\mathbb R} e^{i\lambda\phi(x,t)}a(x,t) f(t)dt\end{displaymath}

where $\lambda >1$, $a\in C_c^\infty(\mathbb{R} ^n\times \mathbb{R} )$ and $\phi$ is a real-valued function in $C^\infty(\mathbb{R} ^n\times \mathbb{R} )$. This operator may be thought of as a variable-curve version of the adjoint of the Fourier restriction operator for space curves. Under a certain nondegeneracy condition on $\phi$, we obtain $L^p-L^q$ estimates for $T_{\lambda}$ with a suitable bound for the operator norm $\Vert T_\lambda \Vert _{L^p\to L^q}$. This generalizes a result of Hörmander for the plane to higher dimensions.


References [Enhancements On Off] (What's this?)

  • [BO] J.-G. Bak and D. Oberlin, A note on Fourier restriction for curves in $\mathbb{R} ^3$, Proceedings of the AMS Conference on Harmonic Analysis, Mt. Holyoke College (June 2001), Contemp. Math., Vol. 320, Amer. Math. Soc., Providence, RI, 2003.
  • [B] J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris 301 (1985), 499-502. MR 87b:42023
  • [BL] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, New York, 1976. MR 58:2349
  • [C] M. Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), 223-238. MR 87b:42018
  • [CS] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disk, Studia Math. 44 (1972), 287-299. MR 50:14052
  • [CSWW] A. Carbery, A. Seeger, S. Wainger, and J. Wright, Classes of singular integral operators along variable lines, J. Geom. Anal. 9 (1999), 583-605. MR 2001g:42026
  • [D1] S. Drury, Restrictions of Fourier transforms to curves, Ann. Inst. Fourier (Grenoble) 35 (1985), 117-123. MR 86e:42026
  • [D2] S. Drury, Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc. 108 (1990), 89-96. MR 91h:42021
  • [DM1] S. Drury and B. Marshall, Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc. 97 (1985), 111-125. MR 87b:42019
  • [DM2] S. Drury and B. Marshall, Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc. 101 (1987), 541-553. MR 88e:42026
  • [F] C. Feffermann, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. MR 41:2468
  • [G] G. Glaeser. Fonctions composées différentiables, Ann. of Math. 77 (1963), 193-209. MR 26:624
  • [GG] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 49:6269
  • [GS] A. Greenleaf and A. Seeger, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), 1077-1119. MR 99g:58120
  • [H] L. Hörmander, Oscillatory integrals and multipliers on $FL^p$, Ark. Mat. 11 (1973), 1-11. MR 49:5674
  • [M] G. Mockenhaupt, Bounds in Lebesgue spaces of oscillatory integral operators, Habilitationsschrift, Universität Siegen (1996).
  • [MSS] G. Mockenhaupt, A. Seeger, and C. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), 60-130. MR 93h:58150
  • [PS] G. Polya and G. Szegö, Problems and theorems in analysis, Die Grundlehren der mathematischen Wissenschaften, Band 216, Springer-Verlag, New York-Heidelberg, 1976. MR 53:2
  • [P1] E. Prestini, A restriction theorem for space curves, Proc. Amer. Math. Soc. 70 (1978), 8-10. MR 57:7026
  • [P2] E. Prestini, Restriction theorems for the Fourier transform to some manifolds in $R\sp{n}$, Proc. Sympos. Pure Math. 35 (1979), 101-109. MR 81d:42028
  • [S] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002

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Additional Information

Jong-Guk Bak
Affiliation: Pohang University of Science and Technology and The Korea Institute for Advanced Study
Email: bak@postech.ac.kr

Sanghyuk Lee
Affiliation: Pohang University of Science and Technology, Pohang 790-784, Korea
Email: huk@euclid.postech.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-03-07144-2
Keywords: Oscillatory integral, restriction theorem
Received by editor(s): October 15, 2002
Received by editor(s) in revised form: December 16, 2002
Published electronically: December 5, 2003
Additional Notes: Research supported in part by KOSEF grant 1999-2-102-003-5
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society

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