Local smoothing of Fourier integral operators and Carleson-Sjölin estimates

Mockenhaupt, Gerd; Seeger, Andreas; Sogge, Christopher D.

doi:10.1090/S0894-0347-1993-1168960-6

Local smoothing of Fourier integral operators and Carleson-Sjölin estimates
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by Gerd Mockenhaupt, Andreas Seeger and Christopher D. Sogge

J. Amer. Math. Soc. 6 (1993), 65-130

DOI: https://doi.org/10.1090/S0894-0347-1993-1168960-6

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References

Richard Beals, Spatially inhomogeneous pseudodifferential operators. II, Comm. Pure Appl. Math. 27 (1974), 161–205. MR 467397, DOI 10.1002/cpa.3160270204
Pierre H. Bérard, Riesz means of Riemannian manifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 1–12. MR 573426
Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885, DOI 10.1007/978-3-642-61876-5
J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 874045, DOI 10.1007/BF02792533
J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
Philip Brenner, $L_{p}-L_{p’}$-estimates for Fourier integral operators related to hyperbolic equations, Math. Z. 152 (1977), no. 3, 273–286. MR 430872, DOI 10.1007/BF01488969
Anthony Carbery, The boundedness of the maximal Bochner-Riesz operator on $L^{4}(\textbf {R}^{2})$, Duke Math. J. 50 (1983), no. 2, 409–416. MR 705033
Anthony Carbery, George Gasper, and Walter Trebels, Radial Fourier multipliers of $L^{p}(\textbf {R}^{2})$, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3254–3255. MR 747595, DOI 10.1073/pnas.81.10.3254
Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. (errata insert). MR 361607, DOI 10.4064/sm-44-3-287-299
Michael Christ, On almost everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985), no. 1, 16–20. MR 796439, DOI 10.1090/S0002-9939-1985-0796439-7
Yves Colin de Verdière, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv. 54 (1979), no. 3, 508–522 (French). MR 543346, DOI 10.1007/BF02566290
A. Córdoba, A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), no. 3, 505–511. MR 544242, DOI 10.1215/S0012-7094-79-04625-8
Antonio Córdoba, Geometric Fourier analysis, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vii, 215–226 (English, with French summary). MR 688026
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR 405514, DOI 10.1007/BF01405172
Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 320624, DOI 10.1007/BF02771772
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Allan Greenleaf and Gunther Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal. 89 (1990), no. 1, 202–232. MR 1040963, DOI 10.1016/0022-1236(90)90011-9
Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 609014, DOI 10.1007/BF02391913
Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052

The analysis of linear partial differential operators, Vols

Oscillatory integrals and multipliers on

11

J.-L. Journé, A. Soffer, and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math. 44 (1991), no. 5, 573–604. MR 1105875, DOI 10.1002/cpa.3160440504
Makoto Kaneko and Gen-ichirô Sunouchi, On the Littlewood-Paley and Marcinkiewicz functions in higher dimensions, Tohoku Math. J. (2) 37 (1985), no. 3, 343–365. MR 799527, DOI 10.2748/tmj/1178228647
Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Wave front sets, local smoothing and Bourgain’s circular maximal theorem, Ann. of Math. (2) 136 (1992), no. 1, 207–218. MR 1173929, DOI 10.2307/2946549
Andreas Seeger, On quasiradial Fourier multipliers and their maximal functions, J. Reine Angew. Math. 370 (1986), 61–73. MR 852510, DOI 10.1515/crll.1986.370.61
A. Seeger and C. D. Sogge, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J. 59 (1989), no. 3, 709–736. MR 1046745, DOI 10.1215/S0012-7094-89-05932-2
A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), no. 3, 669–682. MR 1017329, DOI 10.1512/iumj.1989.38.38031
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
Christopher D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), no. 1, 43–65. MR 835795, DOI 10.1215/S0012-7094-86-05303-2
Christopher D. Sogge, Concerning the $L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123–138. MR 930395, DOI 10.1016/0022-1236(88)90081-X
Christopher D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. (2) 126 (1987), no. 2, 439–447. MR 908154, DOI 10.2307/1971356
Christopher D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), no. 2, 349–376. MR 1098614, DOI 10.1007/BF01245080
Christopher D. Sogge and Elias M. Stein, Averages over hypersurfaces. II, Invent. Math. 86 (1986), no. 2, 233–242. MR 856844, DOI 10.1007/BF01389070
C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms, J. Analyse Math. 54 (1990), 165–188. MR 1041180, DOI 10.1007/BF02796147
Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174
E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355. MR 864375
Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
Robert S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis 5 (1970), 218–235. MR 0257581, DOI 10.1016/0022-1236(70)90027-3
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
Alan Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 (1977), no. 4, 883–892. MR 482878