Invariant Radon transforms on a symmetric space

Orloff, Jeremy

doi:10.1090/S0002-9947-1990-0958898-2

Invariant Radon transforms on a symmetric space
HTML articles powered by AMS MathViewer

by Jeremy Orloff PDF

Trans. Amer. Math. Soc. 318 (1990), 581-600 Request permission

Abstract:

Injectivity and support theorems are proved for a class of Radon transforms, ${R_\mu }$, for $\mu$ a smooth family of measures defined on a certain space of affine planes in ${\mathbb {X}_0}$, where ${\mathbb {X}_0}$ is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against $\mu$ over these planes. We show that if ${R_\mu }f$ is supported inside a ball of radius $R$ then so is $f$. This is true for $f \in L_c^2({\mathbb {X}_0})$ or $f \in \mathcal {E}’({\mathbb {X}_0})$. Furthermore, ${R_\mu }$ is invertible on either of these domains. The main technique is to use facts about spherical harmonics to reduce the problem to a one-dimensional integral equation.

References

I. M. Gel′fand, M. I. Graev, and Z. Ja. Šapiro, Differential forms and integral geometry, Funkcional. Anal. i Priložen. 3 (1969), no. 2, 24–40 (Russian). MR 0244919
Victor Guillemin, The Radon transform on Zoll surfaces, Advances in Math. 22 (1976), no. 1, 85–119. MR 426063, DOI 10.1016/0001-8708(76)90139-0
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
Sigurđur Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180. MR 172311, DOI 10.1007/BF02391776
Sigurđur Helgason, A duality in integral geometry on symmetric spaces, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 37–56. MR 0229191
Bertram Kostant, On the existence and irreducibility of certain series of representations, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 231–329. MR 0399361
B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, DOI 10.2307/2373470
Eric Todd Quinto, Injectivity of rotation invariant Radon transforms on complex hyperplanes in $\textbf {C}^n$, Integral geometry (Brunswick, Maine, 1984) Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 245–260. MR 876322, DOI 10.1090/conm/063/876322
Eric Todd Quinto, The invertibility of rotation invariant Radon transforms, J. Math. Anal. Appl. 91 (1983), no. 2, 510–522. MR 690884, DOI 10.1016/0022-247X(83)90165-8
Kôsaku Yosida, Lectures on differential and integral equations, Pure and Applied Mathematics, Vol. X, Interscience Publishers, New York-London, 1960. MR 0118869