On continuity of singular Fourier integral operators

Authors:
Andrew Comech and Scipio Cuccagna

Journal:
Trans. Amer. Math. Soc. **355** (2003), 2453-2476

MSC (2000):
Primary 35S30

DOI:
https://doi.org/10.1090/S0002-9947-03-02929-5

Published electronically:
February 7, 2003

MathSciNet review:
1973998

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Abstract | References | Similar Articles | Additional Information

Abstract: We derive continuity of Fourier integral operators with one-sided fold singularities. The argument is based on interpolation of (asymptotics of) estimates and 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 regularity properties of the restrictions of solutions to hyperbolic equations onto timelike hypersurfaces and onto hypersurfaces with characteristic points.

**[Ch95]**M. 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**97e:44007****[Co98]**A. Comech,*Damping estimates for oscillatory integral operators with finite type singularities*, Asymptot. Anal.**18**(1998), 263-278. MR**2000f:42006****[Co99]**A. Comech,*Optimal regularity of Fourier integral operators with one-sided folds*, Comm. Partial Differential Equations**24**(1999), 1263-1281. MR**2000m:35190****[Cu97]**S. Cuccagna,*estimates for averaging operators along curves with two-sided**-fold singularities*, Duke Journal**89**(1997), 203-216. MR**99f:58199****[Ds95]**J. J. Duistermaat,*Fourier integral operators*, Progress in Mathematics, vol. 130, Birkhäuser, Boston, 1995. MR**96m:58245****[GrSe94]**A. Greenleaf and A. Seeger,*Fourier integral operators with fold singularities*, J. Reine Angew. Math.**455**(1994), 35-56. MR**95h:58130****[GrSe98]**A. Greenleaf and A. Seeger,*Fourier integral operators with cusp singularities*, Amer. J. Math.**120**(1998), 1077-1119. MR**99g:58120****[Ho71]**L. Hörmander,*Fourier integral operators I*, Acta Math.**127**(1971), 79-183. MR**52:9299****[MeSj78]**R. B. Melrose and J. Sjöstrand,*Singularities of boundary value problems, I*, Comm. Pure Appl. Math.**31**(1978), 593-617. MR**58:11859****[MeTa85]**R. B. Melrose and M. E. Taylor,*Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle*, Adv. in Math.**55**(1985), 242-315. MR**86m:35095****[PhSt91]**D. H. Phong and E. M. Stein,*Radon transforms and torsion*, Internat. Math. Res. Notices**4**(1991), 49-60. MR**93g:58144****[Se93]**A. Seeger,*Degenerate Fourier integral operators in the plane*, Duke Math. J.**71**(1993), 685-745. MR**94h:35292****[Se98]**A. Seeger,*Radon transforms and finite type conditions*, J. Amer. Math. Soc.**11**(1998), 869-897. MR**99f:58202****[SeSoSt91]**A. Seeger, C. D. Sogge and E. M. Stein,*Regularity properties of Fourier integral operators*, Ann. of Math.**134**(1991), 231-251. MR**92g:35252****[SmSo94]**H. F. Smith and C. D. Sogge,*regularity for the wave equation with strictly convex obstacles*, Duke Math. J.**73**(1994), 97-153. MR**95c:35048****[So93]**C. D. Sogge,*Fourier integrals in classical analysis*, Cambridge University Press, Cambridge, 1993. MR**94c:35178****[Ta98]**D. Tataru,*On the regularity of boundary traces for the wave equation*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),**26**(1998), 185-206. MR**99e:35129**

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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

DOI:
https://doi.org/10.1090/S0002-9947-03-02929-5

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.

Article copyright:
© Copyright 2003
American Mathematical Society