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Radon transforms on affine Grassmannians

Author: Boris Rubin
Journal: Trans. Amer. Math. Soc. 356 (2004), 5045-5070
MSC (2000): Primary 44A12; Secondary 47G10
Published electronically: June 29, 2004
MathSciNet review: 2084410
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Abstract: We develop an analytic approach to the Radon transform $\hat f (\zeta)=\int_{\tau\subset \zeta} f (\tau)$, where $f(\tau)$ is a function on the affine Grassmann manifold $G(n,k)$ of $k$-dimensional planes in $\mathbb{R}^n$, and $\zeta$ is a $k'$-dimensional plane in the similar manifold $G(n,k'), \; k'>k$. For $f \in L^p (G(n,k))$, we prove that this transform is finite almost everywhere on $G(n,k')$ if and only if $\; 1 \le p < (n-k)/(k' -k)$, and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of $\mathbb{R}^{n+1}$. It is proved that the dual Radon transform can be explicitly inverted for $ k+k' \ge n-1$, and interpreted as a direct, ``quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if $ k+k' = n-1$. The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.

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Additional Information

Boris Rubin
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Keywords: Radon transforms, Grassmann manifolds, inversion formulas
Received by editor(s): May 13, 2003
Received by editor(s) in revised form: September 11, 2003
Published electronically: June 29, 2004
Additional Notes: This work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
Dedicated: Dedicated to Professor Lawrence Zalcman on the occasion of his 60th birthday
Article copyright: © Copyright 2004 American Mathematical Society

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