Invariant Radon transforms on a symmetric space

Injectivity and support theorems are proved for a class of Radon transforms, R', ,for ,u a smooth family of measures defined on a certain space of affine planes in X0 , where X0 is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against 8 over these planes. We show that if R,I f is supported inside a ball of radius R then so is f . This is true for f e L2(X0) or f E X'(X0). Furthermore, 1 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. 0. INTRODUCTION Radon transfortns have been extensively studied. See, for example, Guillemin [2], Helgason [6], and Gelfand [1]. In this paper we will generalize and unify several results on Radon transforms. If a compact group K acts on a real vector space X0, then one has the following notion of a K-invariant Radon transform. The ingredients are =0, the space of affine hyperplanes, F = {(x, 4)Ix E } c x 60, and a smooth function uL on F. The Radon transform is defined for f E Lc(X0) by R,lf(4) = fxEf (X)u(X, 4) dx. (dx is Lebesgue measure.) If RP intertwines the actions of K it is called K invariant. This is easily seen to be equivalent to the condition u(k x, k ) = (x, ,) for all k E K and x e 4. It is natural to ask if R is invertible and to look for support theorems. Quinto [11] covers the case when K = 0(n) and X0 = RDn . He proves a support theorem and invertibility for K-invariant Radon transforms. Also, Helgason [3] proves support and injectivity theorems in the case ,u _=1 . Helgason [5] proves a support theorem for the classical Radon transform on Euclidean space. If X0 has a natural complex or quaternionic structure one can play the same game with EWF, the space of complex or quaternionic hyperplanes. (F is either the complex numbers C or the quaternions H.) Quinto [10] covers the case when K = U(n) and X0 = Cn . Once again he proves injectivity and a support theorem. In this paper we will take X0 to be the tangent space of a rank one RieReceived by the editors July 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 44A15; Secondary 43A85. ? 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page