Radon transforms on homogeneous spaces and their application to harmonic analysis

2009 Fiscal Year Final Research Report

• Project/Area Number: 19540208
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: University of Tsukuba
• Principal Investigator: KAKEHI Tomoyuki University of Tsukuba, 大学院・数理物質科学研究科, 准教授 (70231248)
• Co-Investigator(Kenkyū-buntansha): ISOZAKI Hiroshi 筑波大学, 大学院・数理物質科学研究科, 教授 (90111913) ; TAKEUCHI Kiyoshi 筑波大学, 大学院・数理物質科学研究科, 准教授 (70281160) ; KINOXHITA Tamotsu 筑波大学, 大学院・数理物質科学研究科, 准教授 (90301077)
• Project Period (FY): 2007 – 2009
• Keywords: ラドン変換 / 等質空間 / 対称空間 / 像の特徴付け / 反転公式 / シュレディンガー方程式 / 基本解

Research Abstract

We studied Radon transforms on homogeneous spaces, and their applications to harmonic analysis. Our results are as follows. (1) We proved that the ranges of generalized matrix Radon transforms are characterized by Pfaffian type invariant differential operators. In addition, we also obtained the inversion formulas. (2) We proved that under some condition the support of the fundamental solution to the Schroedinger equation on a compact symmetric space becomes a lower dimensional subset at a rational time and that it coincides with the whole symmetric space at an irrational time.

Research Products

• [Journal Article] Pseudodifferentialmulti-product representation of the solution operator of a parabolic equation (2009)
  • Author(s): Isozaki Hiroshi, Le Rousseau Jerome
  • Journal Title: Comm.Partial Differential Equations 34, no. 7-9 Pages: 625-655
  • Peer Reviewed
• [Journal Article] Topological Radon transforms and degree formulas for dual varieties (2008)
  • Author(s): Matsui Yutaka, Takeuchi Kiyoshi
  • Journal Title: Proc.Amer.Math.Soc. 136, no.7 Pages: 2365-2373
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for wave equations with time-dependent coefficients (2008)
  • Author(s): Kinoshita Tamotu, Yagdjian Karen
  • Journal Title: Int.J.Appl.Math.Stat. 13, No. S08 Pages: 1-20
  • Peer Reviewed
• [Journal Article] Uhlmann Gunter Probing for electrical inclusions with complex spherical waves (2007)
  • Author(s): Ide Takanori, Isozaki Hiroshi, Nakata Susumu, Siltanen Samuli
  • Journal Title: Comm.Pure Appl.Math. 60, no. 10 Pages: 1415-1442
  • Peer Reviewed
• [Journal Article] Fundamental solution to the Schroedinger equation on a compact symmetric space
  • Author(s): Kakehi Tomoyuki
  • Journal Title: 数理解析研究所講究録 (掲載確定)
• [Presentation] 「Generalized Matrix Radon transform」 Workshop on Integral Geometry and Group Representations (2009)
  • Author(s): Kakehi Tomoyuki
  • Organizer: Tambara Institute of mathematical Sciences
  • Place of Presentation: the University of Tokyo (Tambara, Numata, Gumma)
  • Year and Date: 20090805-20090810
• [Presentation] Fundamental solution to the Schroedinger equation on a compact symmetric space (2008)
  • Author(s): Kakehi Tomoyuki
  • Organizer: Special Session on Radon Transforms, Tomography, and Related Geometric Analysis 2008 AMS Spring Southeastern Meeting (AMS Sectional Meeting)
  • Place of Presentation: Louisiana State University, Baton Rouge, USA
  • Year and Date: 20080328-20080330
• [Presentation] Some remarks on the Schroedinger equation on compact symmetric spaces (2007)
  • Author(s): Kakehi Tomoyuki
  • Organizer: International Conference on Integral Geometry, Harmonic Analysis and Representation Theory
  • Place of Presentation: University of Iceland, Natural Sciences Building, Reykjavik, Iceland
  • Year and Date: 20070815-20070818