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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: December 5, 2003
MathSciNet review: 2053345
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-03-07144-2
PII: S 0002-9939(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