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$L^p\to L^q$ 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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  • [AGZV88] V. I. Arnol'd, S. M. Guse{\u{\i}}\kern.15emn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Birkhäuser Boston Inc., Boston, MA, 1988. MR 89g:58024
  • [Bre75] Philip Brenner, On ${L}\sb{p}-{L}\sb{p^{\prime} }$ estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251-254. MR 52:8658
  • [Bre77] -, ${L}\sb{p}-{L}\sb{p'}$-estimates for Fourier integral operators related to hyperbolic equations, Math. Z. 152 (1977), no. 3, 273-286. MR 55:3877
  • [CC03] Andrew Comech and Scipio Cuccagna, On ${L}^p$ continuity of singular Fourier integral operators, Trans. Amer. Math. Soc. 355 (2003), no. 6, 2453-2476.
  • [CCW99] Anthony Carbery, Michael Christ, and James Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), no. 4, 981-1015. MR 2000h:42010
  • [CdV77] Y. Colin de Verdière, Nombre de points entiers dans une famille homothétique de domains de ${\mathbf {r}}$, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 559-575. MR 58:563
  • [Dui74] J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974), 207-281. MR 53:9306
  • [Dui96] -, Fourier integral operators, Birkhäuser Boston Inc., Boston, MA, 1996. MR 96m:58245
  • [GS77] Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, American Mathematical Society, Providence, R.I., 1977, Mathematical Surveys, No. 14. MR 58:24404
  • [GS02] Allan Greenleaf and Andreas Seeger, Oscillatory and Fourier integral operators with degenerate canonical relations, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), no. Vol. Extra, 2002, pp. 93-141.
  • [GSW00] Allan Greenleaf, Andreas Seeger, and Stephen Wainger, Estimates for generalized Radon transforms in three and four dimensions, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), Contemp. Math., vol. 251, Amer. Math. Soc., Providence, RI, 2000, pp. 243-254. MR 2001j:58047
  • [Hör71] Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79-183. MR 52:9299
  • [Hör94] -, The analysis of linear partial differential operators. III, Springer-Verlag, Berlin, 1994, Pseudo-differential operators, Corrected reprint of the 1985 original. MR 95h:35255
  • [JMR00] Jean-Luc Joly, Guy Metivier, and Jeffrey Rauch, Caustics for dissipative semilinear oscillations, Mem. Amer. Math. Soc. 144 (2000), no. 685, viii+72. MR 2000i:35115
  • [Lit73] Walter Littman, ${L}^{p}-{L}^{q}$-estimates for singular integral operators arising from hyperbolic equations, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, R.I., 1973, pp. 479-481. MR 50:10909
  • [Lud66] Donald Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math. 19 (1966), 215-250. MR 33:4446
  • [Mag01] Akos Magyar, Estimates for the wave kernel near focal points on compact manifolds, J. Geom. Anal. 11 (2001), no. 1, 119-128. MR 2002d:58035
  • [MT85] Richard B. Melrose and Michael E. Taylor, Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. in Math. 55 (1985), no. 3, 242-315. MR 86m:35095
  • [PS91] D. H. Phong and E. M. Stein, Radon transforms and torsion, Internat. Math. Res. Notices (1991), no. 4, 49-60. MR 93g:58144
  • [See93] Andreas Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), no. 3, 685-745. MR 94h:35292
  • [Sog93] Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge University Press, Cambridge, 1993. MR 94c:35178
  • [SS94] Hart F. Smith and Christopher D. Sogge, ${L}^ p$ regularity for the wave equation with strictly convex obstacles, Duke Math. J. 73 (1994), no. 1, 97-153. MR 95c:35048
  • [SSS91] Andreas Seeger, Christopher D. Sogge, and Elias M. Stein, Regularity properties of Fourier integral operators, Ann. of Math. (2) 134 (1991), no. 2, 231-251. MR 92g:35252
  • [Str70] Robert S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461-471. MR 41:876
  • [Sug94] Mitsuru Sugimoto, A priori estimates for higher order hyperbolic equations, Math. Z. 215 (1994), no. 4, 519-531. MR 95j:35128
  • [Sug96] -, Estimates for hyperbolic equations with non-convex characteristics, Math. Z. 222 (1996), no. 4, 521-531. MR 97f:35122
  • [Sug98] -, Estimates for hyperbolic equations of space dimension 3, J. Funct. Anal. 160 (1998), no. 2, 382-407. MR 2000i:35113
  • [Tom79] Peter A. Tomas, Restriction theorems for the Fourier transform, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 111-114. MR 81d:42029

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Additional Information

Andrew Comech
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708

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

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