仿射Grassmann流形上Radon变换的矩条件和支撑定理
Moment Conditions and Support Theorems for Radon Transforms on Affine Grassmann Manifolds
摘要：
令$G(p,n)$和$G(q,n)$分别是${\mathbb R}^n$中$p$ -平面和$q$ -平面的仿射格拉斯曼流形，并且令$\mathcal{R}^{(p,q)}$是由包含关联关系引起的从$G(p,n)$上的光滑函数到$G(q,n)$上的光滑函数的Radon变换。当$p<q$和$\dim G(p,n) = \dim G(p,n)$同时存在时，利用矩条件给出了$\mathcal{R}^{(p,q)}$的值域刻画定理.然后，我们使用这个范围的结果来证明$\mathcal{R}^{(p,q)}$的支持定理。这补充了以前的范围表征定理$\mathcal{R}^{(p,q)}$通过微分方程时，$\dim G(p,n) < \dim G(p,n)$。我们还提出了一个支持定理在后一种情况下。
英文摘要：
Let $G(p,n)$ and $G(q,n)$ be the affine Grassmann manifolds of $p$- and $q$- planes in ${\mathbb R}^n$, respectively, and let $\mathcal{R}^{(p,q)}$ be the Radon transform from smooth functions on $G(p,n)$ to smooth functions on $G(q,n)$ arising from the inclusion incidence relation. When $p<q$ and $\dim G(p,n) = \dim G(p,n)$, we present a range characterization theorem for $\mathcal{R}^{(p,q)}$ via moment conditions. We then use this range result to prove a support theorem for $\mathcal{R}^{(p,q)}$. This complements a previous range characterization theorem for $\mathcal{R}^{(p,q)}$ via differential equations when $\dim G(p,n) < \dim G(p,n)$. We also present a support theorem in this latter case.
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