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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$L^p\to L^q$ regularity of Fourier integral operators with caustics


Author: Andrew Comech
Journal: Trans. Amer. Math. Soc. 356 (2004), 3429-3454
MSC (2000): Primary 35S30
Published electronically: April 26, 2004
MathSciNet review: 2055740
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Abstract: The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on $X\times Y$). The caustic set $\Sigma(\matheurb{ C})$of the canonical relation is characterized as the set of points where the rank of the projection $\pi:\matheurb{ C}\to X\times Y$is smaller than its maximal value, $\dim(X\times Y)-1$. We derive the $L^ p(Y)\to L^ q(X)$ estimates on Fourier integral operators with caustics of corank $1$(such as caustics of type $A_{m+1}$, $m\in{\mathbb N}$). For the values of $p$ and $q$outside of a certain neighborhood of the line of duality, $q=p'$, the $L^ p\to L^ q$ estimates are proved to be caustics-insensitive.

We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.


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Additional Information

Andrew Comech
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708
Email: comech@math.duke.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03570-6
PII: S 0002-9947(04)03570-6
Received by editor(s): January 22, 2003
Published electronically: April 26, 2004
Additional Notes: This work was supported in part by the NSF under Grants No. 0296036 and 0200880
Article copyright: © Copyright 2004 American Mathematical Society