2021/08/09 by Francesca Bartolucci, Bartolucci, Francesca, Filippo De Mari +3
Mathematics · Medicine · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2108.04338
openalex publication_date 2021/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Radon transform for a dual pair (X,Ξ), where X=G/K is a noncompact symmetric space and Ξ is the space of horocycles of X. We address the unitarization problem that was considered (and solved in some cases) by Helgason, namely the determination of a pseudo-differential operator such that the pre-composition with the Radon transform extends to a unitary operator Q\colon L2(X)→ L_\flat2(Ξ), where L_\flat2(Ξ) is a closed subspace of L2(Ξ) which accounts for the Weyl symmetries. Furthermore, we show that the unitary extension intertwines the quasi-regular representations of G on L2(X) and L_\flat2(Ξ).