Partial differential equations and modulation spaces

• Project/Area Number: 24740092
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Yamagata University
• Principal Investigator: KOBAYASHI Masaharu 山形大学, 理学部, 准教授 (30516480)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: モジュレーション空間 / ウィーナー・アマルガム空間 / 関数空間 / シュレディンガー方程式 / ウィーナーアマルガム空間 / フーリエ・ルベーグ空間 / 波束変換 / 波面集合 / ウィナーアマルガム空間

Outline of Final Research Achievements

We introduce a new representation of the Schr\"odinger operators of a free particle and a harmonic oscillator by using the short time Fourier transform. As applications, we give new estimates for the solution to the Schr\"odinger equation with quadratic and sub-quadratic potentials in the framework of modulation spaces. We also characterize the Fourier-Lebesgue type wave front set by using the wave packet transform.

Research Products

• [Journal Article] Estimates on modulation spaces for Schr\"odinger evolution operators with quadratic and sub-quadratic potentials (2014)
  • Author(s): K.Kato, M.Kobayashi, S.Ito
  • Journal Title: J. Funct. Anal. Volume: 266 Issue: 2 Pages: 733-753
  • DOI: 10.1016/j.jfa.2013.08.017
  • Peer Reviewed
• [Journal Article] Characterization of Wave Front Sets in Fourier-Lebesgue Spaces and Its Application (2013)
  • Author(s): K.Kato, M.Kobayashi, S.Ito
  • Journal Title: Funkcialaj Ekvacioj Volume: 56 Issue: 1 Pages: 1-17
  • DOI: 10.1619/fesi.56.1
  • NAID: 130003363159
  • ISSN: 0532-8721
  • Peer Reviewed
• [Journal Article] REPRESENTATION OF SCHRÖDINGER OPERATOR OF A FREE PARTICLE VIA SHORT-TIME FOURIER TRANSFORM AND ITS APPLICATIONS (2012)
  • Author(s): K.Kato, M.Kobayashi, S. Ito
  • Journal Title: Tohoku Mathematical Journal, Second Series Volume: 64 Issue: 2 Pages: 223-231
  • DOI: 10.2748/tmj/1341249372
  • NAID: 130005128512
  • ISSN: 0040-8735, 2186-585X
  • Peer Reviewed
• [Journal Article] Application of wave packet transform of Schr\"odinger equations (2012)
  • Author(s): K.Kato, S.Ito, M.Kobayashi
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B33 Pages: 29-39
  • Peer Reviewed
• [Journal Article] Remarks on Wiener amalgam space type estimates for Schr\"odinger equation (2012)
  • Author(s): K.Kato, M.Kobayashi, S. Ito
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B33 Pages: 41-48
  • Peer Reviewed
• [Presentation] Modulation空間について (2015)
  • Author(s): 小林政晴
  • Organizer: Saga Workshop on Partial Differential Equations
  • Place of Presentation: 佐賀大学
  • Year and Date: 2015-03-04
  • Invited
• [Presentation] Inclusion relation between Lp-Sobolev and Wiener amalgam spaces (2014)
  • Author(s): M.Kobayashi
  • Organizer: 北京大学セミナー
  • Place of Presentation: 中華人民共和国, 北京大学
  • Year and Date: 2014-10-28
  • Invited
• [Presentation] Modulation spaces and Schr\"odinger equations (2014)
  • Author(s): M.Kobayashi
  • Organizer: 2nd East Asian Conference in Harmonic Analysis and Applications
  • Place of Presentation: 中華人民共和国, 牡丹江師範学院
  • Year and Date: 2014-07-13
• [Presentation] Representation of Schr\"odinger operator of a free particle via short-time FourierTransform (2012)
  • Author(s): K.Kato, M.Kobayashi, S.Ito
  • Organizer: Fourier Analysis and Pseudo-Differential Operators
  • Place of Presentation: Aalto University
  • Invited
• [Presentation] Representation of Schr\"odinger operator of a free particle via short-time Fourier Transform (2012)
  • Author(s): K.Kato, M.Kobayashi, S.Ito
  • Organizer: Harmonic Analysis and its Applications at Tokyo 2012
  • Place of Presentation: 首都大学東京
  • Invited
• [Presentation] The inclusion relation between Sobolev and modulation spaces
  • Author(s): M.Kobayashi
  • Organizer: Report (1)
  • Place of Presentation: 北京師範大学, 中華人民共和国, 北京市
  • Invited
• [Presentation] Representation of Schr\"odinger operator of a free particle via short-time Fourier transform
  • Author(s): M.Kobayashi
  • Organizer: Report (2)
  • Place of Presentation: 北京師範大学, 中華人民共和国, 北京市
  • Invited
• [Presentation] Modulation spaces and Schrodinger equations
  • Author(s): M.Kobayashi
  • Organizer: 第６回名古屋微分方程式研究集会
  • Place of Presentation: 名古屋大学
  • Invited