# On oscillatory integral operators with folding canonical relations

Allan Greenleaf; Andreas Seeger

Studia Mathematica (1999)

- Volume: 132, Issue: 2, page 125-139
- ISSN: 0039-3223

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topGreenleaf, Allan, and Seeger, Andreas. "On oscillatory integral operators with folding canonical relations." Studia Mathematica 132.2 (1999): 125-139. <http://eudml.org/doc/216590>.

@article{Greenleaf1999,

abstract = {Sharp $L^p$ estimates are proven for oscillatory integrals with phase functions Φ(x,y), (x,y) ∈ X × Y, under the assumption that the canonical relation $C_Φ$ projects to T*X and T*Y with fold singularities.},

author = {Greenleaf, Allan, Seeger, Andreas},

journal = {Studia Mathematica},

keywords = {oscillatory integrals; Fourier integral operators; fold singularities; oscillatory integral; canonical relation; fold singularity},

language = {eng},

number = {2},

pages = {125-139},

title = {On oscillatory integral operators with folding canonical relations},

url = {http://eudml.org/doc/216590},

volume = {132},

year = {1999},

}

TY - JOUR

AU - Greenleaf, Allan

AU - Seeger, Andreas

TI - On oscillatory integral operators with folding canonical relations

JO - Studia Mathematica

PY - 1999

VL - 132

IS - 2

SP - 125

EP - 139

AB - Sharp $L^p$ estimates are proven for oscillatory integrals with phase functions Φ(x,y), (x,y) ∈ X × Y, under the assumption that the canonical relation $C_Φ$ projects to T*X and T*Y with fold singularities.

LA - eng

KW - oscillatory integrals; Fourier integral operators; fold singularities; oscillatory integral; canonical relation; fold singularity

UR - http://eudml.org/doc/216590

ER -

## References

top- [1] J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 499-502.
- [2] A. P. Calderón and R. Vaillancourt, A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 1185-1187. Zbl0244.35074
- [3] A. Carbery, A. Seeger, S. Wainger and J. Wright, Classes of singular integral operators along variable lines, J. Geom. Anal., to appear. Zbl0964.42003
- [4] M. Christ, Failure of an endpoint estimate for integrals along curves, in: Fourier Analysis and Partial Differential Equations, J. García-Cuerva, E. Hernandez, F. Soria and J. L. Torrea (eds.), Stud. Adv. Math., CRC Press, 1995, 163-168. Zbl0871.42017
- [5] A. Comech, Oscillatory integral operators in scattering theory, Comm. Partial Differential Equations 22 (1997), 841-867. Zbl0883.35142
- [6] S. Cuccagna, ${L}^{2}$ estimates for averaging operators along curves with two-sided k-fold singularities, Duke Math. J. 89 (1997), 203-216. Zbl0908.47050
- [7] A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35-56. Zbl0799.42008
- [8] A. Greenleaf and A. Seeger, Fourier integral operators with simple cusps, Amer. J. Math., to appear.
- [9] A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudo-differential operators with singular symbols, J. Funct. Anal. 89 (1990), 202-232. Zbl0717.44001
- [10] L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79-183. Zbl0212.46601
- [11] L. Hörmander, Oscillatory integrals and multipliers on $F{L}^{p}$, Ark. Mat. 11 (1973), 1-11. Zbl0254.42010
- [12] R. Melrose and M. Taylor, Near peak scattering and the correct Kirchhoff approximation for a convex obstacle, Adv. Math. 55 (1985), 242-315. Zbl0591.58034
- [13] Y. Pan, Hardy spaces and oscillatory integral operators, Rev. Mat. Iberoamericana 7 (1991), 55-64.
- [14] Y. Pan, Hardy spaces and oscillatory integral operators, II, Pacific J. Math. 168 (1995), 167-182.
- [15] Y. Pan and C. D. Sogge, Oscillatory integrals associated to folding canonical relations, Colloq. Math. 61 (1990), 413-419. Zbl0746.58076
- [16] D. H. Phong, Singular integrals and Fourier integral operators, in: Essays on Fourier Analysis in Honor of Elias M. Stein (C. Fefferman, R. Fefferman and S. Wainger, eds.), Princeton Math. Ser. 42, Princeton Univ. Press, 1995, 286-320.
- [17] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157. Zbl0622.42011
- [18] D. H. Phong and E. M. Stein, Radon transforms and torsion, Internat. Math. Res. Notices 4 (1991), 49-60. Zbl0761.46033
- [19] D. H. Phong and E. M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. 140 (1994), 703-722. Zbl0833.43004
- [20] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals, J. Funct. Anal. 73 (1987), 179-194. Zbl0622.42010
- [21] A. Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), 685-745. Zbl0806.35191
- [22] H. Smith and C. D. Sogge, ${L}^{p}$ regularity for the wave equation with strictly convex obstacles, Duke Math. J. 73 (1994), 97-153. Zbl0805.35169
- [23] E. M. Stein, Oscillatory integrals in Fourier analysis, in: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1986, 307-356.
- [24] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. Zbl0821.42001
- [25] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. Zbl0232.42007

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