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

1.仿射格拉斯曼流形上的普法夫型算子和Radon变换：令G（d，n）是R^n中所有d维平面的仿射格拉斯曼流形。定义Radon变换R^p_q为G（p，n）上的光滑函数到G（q，n）上的光滑函数的变换，该变换由包含关联关系产生.然后，我们的结果陈述如下。(1)在p < q的情况下设s和r分别是G（p，n）的秩. G（q，n））。我们假设S < R。然后刻画了R^p_q的值域为一个2s + 2阶Pfavian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。然后给出了R^p_q的反演公式：DR^p_qR^p_q = I，其中D是由Pfidian型算子组成的再生算子. (3)在p > q的情况下。我们假设S < R。然后将R^p_q的值域刻画为一个s + 1阶不变微分方程组的核，该方程组由两种不同类型的Pfrons构成.这项研究是与F。冈萨雷斯。Radon变换的Sobolev估计：基本上Radon变换是函数在子流形上的积分。因此，可以预期拉冬变换在某种程度上正则化了一个函数，事实上，哈茨证明了q平面变换R^0_4将L^2上的一个函数映射到H^<（9）/（2）>上的一个函数，即9/2阶的索伯列夫空间。在这种情况下，规则性的增益与相应的双纤维化的纤维的尺寸成比例。然而，在R^p_q对一般p和q的情况下，我们发现R^p_q并没有正则化函数，以至于正则性的增益不再与纤维的维数成正比。

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.