On oscillatory integral operators with folding canonical relations

Greenleaf, Allan; Seeger, Andreas

Studia Mathematica, Tome 133 (1999), p. 125-139 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 1999-01-01

Zbl 0922.35194

EUDML-ID : urn:eudml:doc:216590

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     author = {Allan Greenleaf and Andreas Seeger},
     title = {On oscillatory integral operators with folding canonical relations},
     journal = {Studia Mathematica},
     volume = {133},
     year = {1999},
     pages = {125-139},
     zbl = {0922.35194},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv132i2p125bwm}
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Greenleaf, Allan; Seeger, Andreas. On oscillatory integral operators with folding canonical relations. Studia Mathematica, Tome 133 (1999) pp. 125-139. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv132i2p125bwm/

Bibliographie

[00000] [1] J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 499-502.

[00001] [2] A. P. Calderón and R. Vaillancourt, A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 1185-1187. | Zbl 0244.35074

[00002] [3] A. Carbery, A. Seeger, S. Wainger and J. Wright, Classes of singular integral operators along variable lines, J. Geom. Anal., to appear. | Zbl 0964.42003

[00003] [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. | Zbl 0871.42017

[00004] [5] A. Comech, Oscillatory integral operators in scattering theory, Comm. Partial Differential Equations 22 (1997), 841-867. | Zbl 0883.35142

[00005] [6] S. Cuccagna, $L^{2}$ estimates for averaging operators along curves with two-sided k-fold singularities, Duke Math. J. 89 (1997), 203-216. | Zbl 0908.47050

[00006] [7] A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35-56. | Zbl 0799.42008

[00007] [8] A. Greenleaf and A. Seeger, Fourier integral operators with simple cusps, Amer. J. Math., to appear.

[00008] [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. | Zbl 0717.44001

[00009] [10] L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79-183. | Zbl 0212.46601

[00010] [11] L. Hörmander, Oscillatory integrals and multipliers on $F L^{p}$ , Ark. Mat. 11 (1973), 1-11. | Zbl 0254.42010

[00011] [12] R. Melrose and M. Taylor, Near peak scattering and the correct Kirchhoff approximation for a convex obstacle, Adv. Math. 55 (1985), 242-315. | Zbl 0591.58034

[00012] [13] Y. Pan, Hardy spaces and oscillatory integral operators, Rev. Mat. Iberoamericana 7 (1991), 55-64.

[00013] [14] Y. Pan, Hardy spaces and oscillatory integral operators, II, Pacific J. Math. 168 (1995), 167-182.

[00014] [15] Y. Pan and C. D. Sogge, Oscillatory integrals associated to folding canonical relations, Colloq. Math. 61 (1990), 413-419. | Zbl 0746.58076

[00015] [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.

[00016] [17] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157. | Zbl 0622.42011

[00017] [18] D. H. Phong and E. M. Stein, Radon transforms and torsion, Internat. Math. Res. Notices 4 (1991), 49-60. | Zbl 0761.46033

[00018] [19] D. H. Phong and E. M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. 140 (1994), 703-722. | Zbl 0833.43004

[00019] [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. | Zbl 0622.42010

[00020] [21] A. Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), 685-745. | Zbl 0806.35191

[00021] [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. | Zbl 0805.35169

[00022] [23] E. M. Stein, Oscillatory integrals in Fourier analysis, in: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1986, 307-356.

[00023] [24] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. | Zbl 0821.42001

[00024] [25] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. | Zbl 0232.42007