^{1} Department of Mathematical Sciences, Faculty of Science, Københavns Universitet^{2} Department of Mathematical Sciences, Faculty of Science, Københavns Universitet

DOI:

10.1016/j.aim.2013.03.010

Abstract:

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms. A reductive symmetric space is a homogeneous space G/H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ(P) is opposite to P and let NP be the unipotent radical of P. For a compactly supported smooth function ϕ on G/H, we define RP(ϕ)(g) to be the integral of NP∋n↦ϕ(gn⋅H) over NP. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions. For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of RPϕ. The proof is based on the relation between the Radon transform and the Fourier transform on G/H, and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

Type:

Journal article

Language:

English

Published in:

Advances in Mathematics, 2013, Vol 240, p. 427-483