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

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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.