On X-ray transforms for rigid line complexes and integrals over curves in $\mathbb {R}^4$
HTML articles powered by AMS MathViewer
- by Allan Greenleaf, Andreas Seeger and Stephen Wainger PDF
- Proc. Amer. Math. Soc. 127 (1999), 3533-3545 Request permission
Abstract:
Endpoint estimates are proved for model cases of restricted X-ray transforms and singular fractional integral operators in $\mathbb {R}^{4}$.References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Jean Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 10, 499–502 (French, with English summary). MR 812567
- Michael Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), no. 1, 223–238. MR 766216, DOI 10.1090/S0002-9947-1985-0766216-6
- —, Endpoint bounds for singular fractional integral operators, preprint, unpublished (1988).
- I. M. Gel′fand and M. I. Graev, Line complexes in the space $C^{n}$, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 39–52 (Russian). MR 0238246
- 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
- —, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), 1077–1119.
- Allan Greenleaf and Gunther Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240. MR 1016420, DOI 10.1215/S0012-7094-89-05811-0
- Walter Littman, $L^{p}-L^{q}$-estimates for singular integral operators arising from hyperbolic equations, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 479–481. MR 0358443
- David McMichael, Damping oscillatory integrals with polynomial phase, Math. Scand. 73 (1993), no. 2, 215–228. MR 1269260, DOI 10.7146/math.scand.a-12467
- 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
- A. Nagel, E. M. Stein, and S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 3, 1060–1062. MR 466470, DOI 10.1073/pnas.75.3.1060
- Daniel M. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), no. 1, 56–60. MR 866429, DOI 10.1090/S0002-9939-1987-0866429-6
- Daniel M. Oberlin, Oscillatory integrals with polynomial phase, Math. Scand. 69 (1991), no. 1, 45–56. MR 1143473, DOI 10.7146/math.scand.a-12368
- Daniel M. Oberlin, A convolution estimate for a measure on a curve in $\mathbf R^4$, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1355–1361. MR 1363436, DOI 10.1090/S0002-9939-97-03716-7
- —, An estimate for a restricted X-ray transform, preprint.
- Fulvio Ricci and Elias M. Stein, Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds, J. Funct. Anal. 86 (1989), no. 2, 360–389. MR 1021141, DOI 10.1016/0022-1236(89)90057-8
- S. Secco, Fractional integration along homogeneous curves in $\mathbb {R}^{3}$, preprint .
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Robert S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471. MR 256219, DOI 10.1090/S0002-9947-1970-0256219-1
Additional Information
- Allan Greenleaf
- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
- Email: allan@math.rochester.edu
- Andreas Seeger
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 226036
- Email: seeger@math.wisc.edu
- Stephen Wainger
- MR Author ID: 179960
- Email: wainger@math.wisc.edu
- Received by editor(s): January 13, 1998
- Published electronically: August 5, 1999
- Additional Notes: This research was supported in part by grants from the National Science Foundation.
- Communicated by: Christopher D. Sogge
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3533-3545
- MSC (1991): Primary 44A12; Secondary 35S30
- DOI: https://doi.org/10.1090/S0002-9939-99-05379-4
- MathSciNet review: 1670367