Dual Radon transforms on affine Grassmann manifolds

Gonzalez, Fulton; Kakehi, Tomoyuki

doi:10.1090/S0002-9947-04-03471-3

Dual Radon transforms on affine Grassmann manifolds
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by Fulton B. Gonzalez and Tomoyuki Kakehi

Trans. Amer. Math. Soc. 356 (2004), 4161-4180

DOI: https://doi.org/10.1090/S0002-9947-04-03471-3

Published electronically: April 16, 2004

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Abstract:

Fix $0 \leq p < q \leq n-1$, and let $G(p,n)$ and $G(q,n)$ denote the affine Grassmann manifolds of $p$- and $q$-planes in $\mathbb {R}^n$. We investigate the Radon transform $\mathcal {R}^{(q,p)} : C^{\infty } (G(q,n)) \to C^{\infty } (G(p,n))$ associated with the inclusion incidence relation. For the generic case $\dim G(q,n) < \dim G(p,n)$ and $p+q > n$, we will show that the range of this transform is given by smooth functions on $G(p,n)$ annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case $p+q =n$.

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