2002 Fiscal Year Final Research Report Summary
Global properties of differential operators of subdeterminantal type and integral geometry on symmetric spaces
| Project/Area Number |
13640203
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | University of Tsukuba |
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Principal Investigator |
KAKEHI Tomoyuki University of Tsukuba, Institute of Mathematics, Associate Professor, 数学系, 助教授 (70231248)
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| Co-Investigator(Kenkyū-buntansha) |
TAIRA Kazuaki University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (90016163)
SASAKI Tateaki University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (80087436)
KAJITANI Kunihiko University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (00026262)
NAITO Satoshi University of Tsukuba, Institute of Mathematics, Associate Professor, 数学系, 助教授 (60252160)
MIYAMOTO Masahiko University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (30125356)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Radon transform / integral geometry / symmetric space / range characterization / inversion formula |
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| Research Abstract |
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.
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