Radon transform on symmetric matrix domains

2007/11/09 by Zhang, Genkai
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0711.1476

Abstract

Let \bbK=\mathbb R, \mathbb C, \mathbb H be the field of real, complex or quaternionic numbers and M_{p, q}(\bbK) the vector space of all p× q-matrices. Let X be the matrix unit ball in M_{n-r, r}(\bbK) consisting of contractive matrices. As a symmetric space, X=G/K=O(n-r, r)/O(n-r)× O(r), U(n-r, r)/U(n-r)× U(r) and respectively Sp(n-r, r)/Sp(n-r)× Sp(r). The matrix unit ball y₀ in M_{r^′-r, r} with r^′ ≤ n-1 is a totally geodesic submanifold of X and let Y be the set of all G-translations of the submanifold y₀. The set Y is then a manifold and an affine symmetric space. We consider the Radon transform \mathcal Rf(y) for functions f∈ C₀^∞(X) defined by integration of f over the subset y, and the dual transform \mathcal R^t F(x), x∈ X for functions F(y) on Y. We find inversion formulas by constructing explicit certain invariant differential operators.

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