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Transactions of the American Mathematical Society

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Radon transform on symmetric matrix domains

Author: Genkai Zhang
Journal: Trans. Amer. Math. Soc. 361 (2009), 1351-1369
MSC (2000): Primary 22E45, 33C67, 43A85, 44A12
Published electronically: October 17, 2008
MathSciNet review: 2457402
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Abstract: Let $ \mathbb{K}=\mathbb{R}, \mathbb{C}, \mathbb{H}$ be the field of real, complex or quaternionic numbers and $ M_{p, q}(\mathbb{K})$ the vector space of all $ p\times q$-matrices. Let $ X$ be the matrix unit ball in $ M_{n-r, r}(\mathbb{K})$ consisting of contractive matrices. As a symmetric space, $ X=G/K=O(n-r, r)/O(n-r)\times O(r)$, $ U(n-r, r)/U(n-r)\times U(r)$ and respectively $ Sp(n-r, r)/Sp(n-r)\times Sp(r)$. The matrix unit ball $ y_0$ in $ M_{r^\prime-r, r}$ with $ r^\prime \le n-1$ is a totally geodesic submanifold of $ X$ and let $ Y$ be the set of all $ G$-translations of the submanifold $ y_0$. The set $ Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $ \mathcal Rf(y)$ for functions $ f\in C_0^\infty(X)$ defined by integration of $ f$ over the subset $ y$, and the dual transform $ \mathcal R^t F(x), x\in X$ for functions $ F(y)$ on $ Y$. For $ 2r <n, 2r\le r^\prime$ with a certain evenness condition in the case $ \mathbb{K}=\mathbb{R}$, we find a $ G$-invariant differential operator $ \mathcal M$ and prove it is the right inverse of $ \mathcal R^t \mathcal R$, $ \mathcal R^t \mathcal R \mathcal M f=c f$, for $ f\in C_0^\infty(X)$, $ c\ne 0$. The operator $ f\to \mathcal R^t\mathcal Rf$ is an integration of $ f$ against a (singular) function determined by the root systems of $ X$ and $ y_0$. We study the analytic continuation of the powers of the function and we find a Bernstein-Sato type formula generalizing earlier work of the author in the set up of the Berezin transform. When $ X$ is a rank one domain of hyperbolic balls in $ \mathbb{K}^{n-1}$ and $ y_0$ is the hyperbolic ball in $ \mathbb{K}^{r^\prime -1}$, $ 1<r^\prime<n$ we obtain an inversion formula for the Radon transform, namely $ \mathcal M\mathcal R^t\mathcal R f=c f$. This generalizes earlier results of Helgason for non-compact rank one symmetric spaces for the case $ r^\prime=n-1$.

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Additional Information

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, Sweden

Keywords: Radon transform, inverse Radon transform, symmetric domains, Grassmannian manifolds, Lie groups, fractional integrations, Bernstein-Sato formula, Cherednik operators, invariant differential operators
Received by editor(s): January 17, 2007
Published electronically: October 17, 2008
Additional Notes: This research was supported by the Swedish Science Council (VR)
Article copyright: © Copyright 2008 American Mathematical Society

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