On $L^{p}$ continuity of singular Fourier integral operators
HTML articles powered by AMS MathViewer
- by Andrew Comech and Scipio Cuccagna PDF
- Trans. Amer. Math. Soc. 355 (2003), 2453-2476 Request permission
Abstract:
We derive $L^{p}$ continuity of Fourier integral operators with one-sided fold singularities. The argument is based on interpolation of (asymptotics of) $L^{2}$ estimates and $\mathrm {H}^1\to L^1$ estimates. We derive the latter estimates elaborating arguments of Seeger, Sogge, and Stein’s 1991 paper. We apply our results to the study of the $L^{p}$ regularity properties of the restrictions of solutions to hyperbolic equations onto timelike hypersurfaces and onto hypersurfaces with characteristic points.References
- Michael Christ, Failure of an endpoint estimate for integrals along curves, Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992) Stud. Adv. Math., CRC, Boca Raton, FL, 1995, pp. 163–168. MR 1330238
- Andrew Comech, Damping estimates for oscillatory integral operators with finite type singularities, Asymptot. Anal. 18 (1998), no. 3-4, 263–278. MR 1668946
- Andrew Comech, Optimal regularity of Fourier integral operators with one-sided folds, Comm. Partial Differential Equations 24 (1999), no. 7-8, 1263–1281. MR 1697488, DOI 10.1080/03605309908821465
- Scipio Cuccagna, $L^2$ estimates for averaging operators along curves with two-sided $k$-fold singularities, Duke Math. J. 89 (1997), no. 2, 203–216. MR 1460620, DOI 10.1215/S0012-7094-97-08910-9
- J. J. Duistermaat, Fourier integral operators, Progress in Mathematics, vol. 130, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1362544
- Allan Greenleaf and Andreas Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35–56. MR 1293873, DOI 10.1515/crll.1994.455.35
- Allan Greenleaf and Andreas Seeger, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), no. 5, 1077–1119. MR 1646055, DOI 10.1353/ajm.1998.0036
- Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
- R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. I, Comm. Pure Appl. Math. 31 (1978), no. 5, 593–617. MR 492794, DOI 10.1002/cpa.3160310504
- 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 778964, DOI 10.1016/0001-8708(85)90093-3
- D. H. Phong and E. M. Stein, Radon transforms and torsion, Internat. Math. Res. Notices 4 (1991), 49–60. MR 1121165, DOI 10.1155/S1073792891000077
- Andreas Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), no. 3, 685–745. MR 1240601, DOI 10.1215/S0012-7094-93-07127-X
- Andreas Seeger, Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (1998), no. 4, 869–897. MR 1623430, DOI 10.1090/S0894-0347-98-00280-X
- 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 1127475, DOI 10.2307/2944346
- 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 1257279, DOI 10.1215/S0012-7094-94-07304-3
- Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579, DOI 10.1017/CBO9780511530029
- Daniel Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 1, 185–206. MR 1633000
Additional Information
- Andrew Comech
- Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599
- Scipio Cuccagna
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
- Received by editor(s): September 4, 1998
- Received by editor(s) in revised form: June 3, 2001
- Published electronically: February 7, 2003
- Additional Notes: Both authors were partially supported by grants from the National Science Foundation.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 2453-2476
- MSC (2000): Primary 35S30
- DOI: https://doi.org/10.1090/S0002-9947-03-02929-5
- MathSciNet review: 1973998