$L^ p$ estimates for the X-ray transform restricted to line complexes of Kirillov type
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- by Hann Tzong Wang PDF
- Trans. Amer. Math. Soc. 332 (1992), 793-821 Request permission
Abstract:
Let there be given a piecewise continuous rectifiable curve $\phi :{\mathbf {R}} \to {{\mathbf {R}}^n}$. Let ${G_{1,n}}({M_{1,n}})$ be the usual Grassmannian (bundle) in ${{\mathbf {R}}^n}$. Define an $n$-dimensional submanifold ${M_\phi }({{\mathbf {R}}^n})$ of ${M_{1,n}}$ as the set of all copies of ${G_{1,n}}$ along the curve $\phi$. Following Kirillov, we know that a nice function $f(x)$ can be recovered from its X-ray transform ${R_{1,n}}f$ on ${M_\phi }({{\mathbf {R}}^n})$ if and only if the curve $\phi$ intersects almost every affine hyperplane. Define a measure on ${M_\phi }({{\mathbf {R}}^n})$ by $d\mu = d{\mu _x}(\pi )d\lambda (x)$, where $d{\mu _x}$ is the probability measure on ${M_{1,n}}$ carried by the set of lines passing through the point $x$ and invariant under the stabilizer of $x$ in $O(n)$ and $d\lambda$ is the usual measure on $\phi$. We show that, if $n > 2$ and $\phi$ is unbounded, then $\left \| {R_{1,n}}f\right \|_{{L^q}({M_\phi }({{\mathbf {R}}^n}),d\mu )} \leq C\left \| f\right \| _{{L^p}({{\mathbf {R}}^n})}$ if and only if $p = q = n - 1$ and $\phi$ is line-like, that is, $\lambda (\phi \cap B(0;R)) = O(R)$. This result gives a classification of Kirillov line complexes in terms of ${L^p}$ estimates.References
- H. J. Brascamp, Elliott H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Functional Analysis 17 (1974), 227–237. MR 0346109, DOI 10.1016/0022-1236(74)90013-5
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- A. P. Calderón, On the Radon transform and some of its generalizations, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 673–689. MR 730101
- 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
- Michael Christ, Estimates for the $k$-plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891–910. MR 763948, DOI 10.1512/iumj.1984.33.33048
- S. W. Drury, Generalizations of Riesz potentials and $L^{p}$ estimates for certain $k$-plane transforms, Illinois J. Math. 28 (1984), no. 3, 495–512. MR 748958, DOI 10.1215/ijm/1256046077
- S. W. Drury, $L^{p}$ estimates for the X-ray transform, Illinois J. Math. 27 (1983), no. 1, 125–129. MR 684547, DOI 10.1215/ijm/1256065417
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79. MR 133102, DOI 10.1112/jlms/s1-37.1.74
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- 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
- Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
- Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
- Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, 300–322. MR 1546052, DOI 10.1215/S0012-7094-38-00423-5
- A. A. Kirillov, A problem of I. M. Gel′fand, Soviet Math. Dokl. 2 (1961), 268–269. MR 0117513
- D. M. Oberlin and E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), no. 5, 641–650. MR 667786, DOI 10.1512/iumj.1982.31.31046
- Kennan T. Smith and Donald C. Solmon, Lower dimensional integrability of $L^{2}$ functions, J. Math. Anal. Appl. 51 (1975), no. 3, 539–549. MR 377496, DOI 10.1016/0022-247X(75)90105-5
- Kennan T. Smith, Donald C. Solmon, and Sheldon L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1227–1270. MR 490032, DOI 10.1090/S0002-9904-1977-14406-6
- 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, $L^p$ estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J. 48 (1981), no. 4, 699–727. MR 782573, DOI 10.1215/S0012-7094-81-04839-0 H. T. Wang, ${L^p}$ estimates for the restricted $X$-ray transform, Ph.D. Dissertation, Univ. of Rochester, 1987.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 332 (1992), 793-821
- MSC: Primary 53C65; Secondary 44A12, 92C55
- DOI: https://doi.org/10.1090/S0002-9947-1992-1052912-6
- MathSciNet review: 1052912