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Global properties of differential operators of subdeterminantal type and integral geometry on symmetric spaces

对称空间上次行列式微分算子与积分几何的全局性质

基本信息

批准号：
13640203
负责人：
KAKEHI Tomoyuki
金额：
$ 2.56万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

1. Pfaffian type operators and Radon transforms on affine Grassmann manifolds : Let G(d, n) be the affine Grassmann manifolds of all d-dimensional planes in R^n". Then the Radon transform R^p_q is defined as the transform from smooth functions on G(p,n) to smooth functions on G(q,n] arising from the inclusion incidence relation. Then our results are stated as follows. (1) In the case p < q. Let s and r be the rank of G(p, n) (resp. G(q, n) ). We assume that s < r. Then the range of R^p_q is characterized as the kernel of a single Pfaffian type invariant differenial operator of order 2s + 2. (2) In the case p < q. We assume that s 【less than or equal】 r. Then the inversion formula for R^p_q is given as DR^p_qR^p_q = I, where D is the reproducing operator consisting of Pfaffian type operators. (3) In the case p > q. We assume that s < r. Then the range of R^p_q is characterized as the kernel of an invariant system of differential equations of order s + 1, which consists of two different kinds of Pfaffians. This research was done in collaboration with F. Gonzalez.2. Sobolev estimates for Radon transforms : Basically a Radon transform is an integration of a function over a submanifold. So it is expected that a Radon transform regularizes a function to some extent, and in fact, it was shown by Strichartz that the q-plane transform R^0_4 maps a function on L^2 to a funtion on H^<(9)/(2)> the Sobolev space of order 9/2. In this case, the gain of regularity is proportional to the demension of the fiber of the corresponding double fibration. However, in the case of R^p_q for general p and q, we discovered that R^p_q does not regularize a function so much in the sense that the gain of regularity is no longer proportional to the dimension of the fiber.

1. 仿射格拉斯曼流形上的普法夫型算子和Radon变换：设G(d, n)为R^n"中所有d维平面的仿射格拉斯曼流形。然后Radon变换R^p_q被定义为由包含关联关系产生的从G(p,n)上的平滑函数到G(q,n]上的平滑函数的变换。那么我们的结果如下。 (1) 在 p < q 的情况下。令 s 和 r 为 G(p, n) （分别为 G(q, n) ）的秩。我们假设 s < r。然后将 R^p_q 的范围表征为 2s + 2 阶单个 Pfaffian 型不变微分算子的核。 (2) 在 p < q 的情况下。我们假设 s 【小于或等于】r。然后给出R^p_q的反演公式 DR^p_qR^p_q = I，其中 D 是由 Pfaffian 类型运算符组成的再现运算符。 (3) 在p>q的情况下。我们假设 s < r。然后将 R^p_q 的范围表征为 s + 1 阶微分方程不变系统的核，该系统由两种不同类型的 Pfaffian 组成。这项研究是与 F. Gonzalez 合作完成的。2。 Sobolev 对 Radon 变换的估计：基本上，Radon 变换是函数在子流形上的积分。因此，期望 Radon 变换在一定程度上正则化函数，事实上，Strichartz 表明 q 平面变换 R^0_4 将 L^2 上的函数映射到 H^<(9)/(2)> 9/2 阶 Sobolev 空间上的函数。在这种情况下，增益为规律性与相应双纤维的纤维尺寸成正比。然而，在 R^p_q 对于一般 p 和 q 的情况下，我们发现 R^p_q 对函数的正则化程度不高，因为正则性增益不再与纤维的尺寸成正比。