Final Draft ON RADON TRANSFORMS ON COMPACT LIE GROUPS
نویسنده
چکیده
We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to S nor to S. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from S.
منابع مشابه
A Method for Proving L-boundedness of Singular Radon Transforms in Codimension 1
Singular Radon transforms are a type of operator combining characteristics of both singular integrals and Radon transforms. They are important in a number of settings in mathematics. In a theorem of M. Christ, A. Nagel, E. Stein, and S. Wainger [3], L boundedness of singular Radon transforms for 1 < p < ∞ is proven under a general finite-type condition using the method of lifting to nilpotent L...
متن کاملCompact Simple Lie Groups and Their C-, S-, and E-Transforms
New continuous group transforms, together with their discretization over a lattice of any density and admissible symmetry, are defined for a general compact simple Lie groups of rank 2 ≤ n < ∞. Rank 1 transforms are known. Rank 2 exposition of Cand S-transforms is in the literature. The E-transforms appear here for the first time.
متن کاملThe Abel, Fourier and Radon transforms on symmetric spaces
In this paper we prove some recent results on the three transforms in the title and discuss their relationships to older results. The spaces we deal with are symmetric spaces X = G/K of the noncompact type, G being a connected noncompact semisimple Lie group with finite center and K a maximal compact subgroup. For the two natural Radon transforms on X we prove a new inversion formula and a shar...
متن کاملDisintegration of Measures on Compact Transformation Groups
To prove 1.1, one first assumes X is compact and G is a Lie group. In this case, X is "measure-theoretically" the product Y x G; this follows from the existence of local cross-sections to the projection n [6]. Let n2 : X ~ Y x G —> G, and define a map £ from L(Y, v) to the space of Radon measures on G as follows: £(ƒ) = TÏ2 [if ° n) ' M] • Apply the Dunford-Pettis Theorem [3] to ? to obtain a m...
متن کاملRadon, Cosine and Sine Transforms on Grassmannian Manifolds
LetGn,r(K) be the Grassmannian manifold of k-dimensionalK-subspaces in K where K = R,C,H is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, Rr′,r, Cr′,r and Sr′,r, from the L space L2(Gn,r(K)) to the space L 2(Gn,r′(K)), for r, r ′ ≤ n − 1. The L spaces are decomposed into irreducible representations of G with multiplicity free. We compute ...
متن کامل