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Potential theory for parabolic equations and the functional analysis

Research Project

Project/Area Number 15K04934
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionOsaka City University

Principal Investigator

Nishio Masaharu  大阪市立大学, 大学院理学研究科, 准教授 (90228156)

Co-Investigator(Kenkyū-buntansha) 下村 勝孝  茨城大学, 理工学研究科(理学野), 教授 (00201559)
竹内 敦司  大阪市立大学, 大学院理学研究科, 准教授 (30336755)
Research Collaborator YAMADA masahiro  
HISHIKAWA yosuke  
TANAKA kiyoki  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsポテンシャル論 / 熱方程式 / 分数ベキラプラシアン / マルチン境界 / ベルグマン空間 / ブロッホ空間 / 調和双対 / 再生核 / Potential 論 / 分数ベキラプラス作用素 / 多重調和関数 / ラプラス方程式 / 多重放物型方程式 / 調和関数 / 再生核ヒルベルト空間 / ベルグマン核
Outline of Final Research Achievements

In the present research, we analysed the heat equation and related more general parabolic equations by using potential theory and the functional analysis. We introduced the analogue of Bergman spaces, Bloch spaces and Hardy spaces, to discuss conjugate functions associated with the parabolic operators. Moreover, we obtained some explicit form of the reproducing kernels, the integral operators of which correspond to the orthogonal projection, for more general function spaces associated with the iterated parabolic operators. As for the heat equation for indefinite metrics, we obtained the Liouville type theorem for conformal mappings, which enables us to investigate caloric morphisms.

Academic Significance and Societal Importance of the Research Achievements

一般的に言って、関数を解析することを考慮したとき、性質の良い、取り扱いやすい関数について詳細に研究調査を行ったのち、近似の手法を用いたアプローチによって目的の結果を得ようと考えることは有効である。そして、本研究の成果はそのような方向に向かって進展しているということができる。すなわち、一般の関数は多重調和関数、あるいは、多重熱方程式の解で近似されることが期待されており、今回、我々の理論が、多重放物型方程式の方向に一般化されてきたことによって、将来、さらなる応用が期待される。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (42 results)

All 2019 2018 2017 2016 2015

All Journal Article (14 results) (of which Peer Reviewed: 11 results,  Acknowledgement Compliant: 3 results) Presentation (28 results) (of which Int'l Joint Research: 8 results,  Invited: 8 results)

  • [Journal Article] Integration by parts formulas for marked Hawkes processes2019

    • Author(s)
      Takeuchi Atsushi
    • Journal Title

      Statistics & Probability Letters

      Volume: 145 Pages: 229-237

    • DOI

      10.1016/j.spl.2018.10.003

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A system of conjugate functions on parabolic Bloch spaces2018

    • Author(s)
      Y.Hishikawa, M.Nishio and M.Yamada
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 70 Issue: 3 Pages: 1085-1102

    • DOI

      10.2969/jmsj/74887488

    • NAID

      130007420340

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remark on pathwise uniqueness of stochastic differential equations driven by L?vy processes2018

    • Author(s)
      Takeuchi Atsushi、Tsukada Hiroshi
    • Journal Title

      Stochastic Analysis and Applications

      Volume: 37 Issue: 2 Pages: 155-170

    • DOI

      10.1080/07362994.2018.1541750

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Harmonic Bergman kernels and Toeplitz operators on the ball with radial measures2017

    • Author(s)
      Nishio, Masaharu; Tanaka, Kiyoki
    • Journal Title

      Rev. Roumaine Math. Pures Appl.

      Volume: 62 Pages: 155-169

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Reproducing kernels for iterated parabolic operators on the upper half space with application to polyharmonic Bergman spaces2017

    • Author(s)
      Nishio, Masaharu; Shimomura, Katsunori
    • Journal Title

      Complex Anal. Oper. Theory

      Volume: 11 Issue: 8 Pages: 1865-1878

    • DOI

      10.1007/s11785-017-0673-5

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Harmonic Bergman kernels and Toeplitz operators on the ball with radial measures2017

    • Author(s)
      Masaharu Nishio and Kiyoki tanaka
    • Journal Title

      Revue Roumaine de Mathemaique Pures et Appliques

      Volume: 62

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A system of conjugate functions on parabolic Bloch spaces2017

    • Author(s)
      Y.Hishikawa; M.Nishio; M.Yamada,
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 印刷中

    • NAID

      130007420340

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Integration by parts formulas for conditional intensities of marked Hawkes processes2017

    • Author(s)
      Atsushi Takeuchi
    • Journal Title

      The Institute of Statistical Mathematics Cooperative Research Report

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
  • [Journal Article] Liouville type theorem on conformal mapping for indefinite metrics associate with ultra-hyperbolic equations2016

    • Author(s)
      Katsunori Shimomura
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 48

    • NAID

      130005429965

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Joint distributions for stochastic functional differential equations2016

    • Author(s)
      Atsushi Takeuchi
    • Journal Title

      Stochastica

      Volume: 88

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] 半空間上の多重調和関数に関する再生核について2016

    • Author(s)
      西尾昌治
    • Journal Title

      数理解析研究所講究録

      Volume: 1980

    • Related Report
      2015 Research-status Report
  • [Journal Article] Joint distributions for stochastic functional differential equations2016

    • Author(s)
      Atsushi Takeuchi
    • Journal Title

      Stochastics: An International Journal of Probability and Stochastic Processes

      Volume: -

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Density of joint distributions for stochastic functional differential equations2016

    • Author(s)
      Atsushi Takeuchi
    • Journal Title

      The Institute of Statistical Mathematics Cooperative Research Report

      Volume: -

    • Related Report
      2015 Research-status Report
  • [Journal Article] Liouville type theorem on conformal mapping for indenite metrics associate with ultra-hyperbolic equations2016

    • Author(s)
      Katsunori Shimomura
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: -

    • NAID

      130005429965

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Presentation] Bergman type spaces for iterated parabolic operators of fractional order2019

    • Author(s)
      Masaharu Nishio
    • Organizer
      Potential Analysis and its Related Fields 2019
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Hilbert spaces and reproducing kernels for parabolic operators of fractional order2018

    • Author(s)
      Masaharu Nishio
    • Organizer
      Second Interdisciplinary and Research Alumni Symposium iJaDe2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Reproducing kernels with respect to function spaces of polyharmonic and polyparabolic functions2018

    • Author(s)
      Masaharu Nishio
    • Organizer
      American Institution Mathematical Sciences
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Caloric morphisms for rotation invariant metrics on semi-euclidean spaces2018

    • Author(s)
      Katsunori Shimomura
    • Organizer
      American Institution Mathematical Sciences
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Bateman変換とCaloric morphism2018

    • Author(s)
      下村勝孝
    • Organizer
      ポテンシャル論研究集会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Density for the solution to stochastic functional differential equations2018

    • Author(s)
      Atsushi Takeuchi
    • Organizer
      Second Interdisciplinary and Research Alumni Symposium iJaDe2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Joint distributions for stochastic functional differential equations2017

    • Author(s)
      竹内敦司
    • Organizer
      Osaka-UCL Workshop on Stochastics, Numerics and Risk
    • Place of Presentation
      大阪大学 (大阪府豊中市)
    • Year and Date
      2017-03-29
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Malliavin calculus for marked Hawkes processes2017

    • Author(s)
      竹内敦司
    • Organizer
      日本数学会2017年度年会
    • Place of Presentation
      首都大学東京 (東京都八王子市)
    • Year and Date
      2017-03-24
    • Related Report
      2016 Research-status Report
  • [Presentation] Hawkes processとMalliavin解析2017

    • Author(s)
      竹内敦司
    • Organizer
      岡山-広島 解析・確率論セミナー2017
    • Place of Presentation
      岡山大学 (岡山県岡山市)
    • Year and Date
      2017-02-20
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 放物型方程式に対するポテンシャル論と関数空間2017

    • Author(s)
      西尾昌治
    • Organizer
      日本数学会 2017 年度秋季総合分科会函数論分科会特別講演
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Weighted polyharmonic and polyparabolic Bergman spaces on the upper half space2017

    • Author(s)
      下村勝孝, 西尾昌治
    • Organizer
      日本数学会 2017 年度秋季総合分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Caloric morphism with Bateman space mapping for radial metrics2017

    • Author(s)
      下村勝孝
    • Organizer
      日本数学会 2017 年度秋季総合分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Integration by parts formula for conditional intensities of marked Hawkes processes2016

    • Author(s)
      竹内敦司
    • Organizer
      統計数理研究所共同研究集会「無限分解可能過程に関連する諸問題」
    • Place of Presentation
      統計数理研究所 (東京都立川市)
    • Year and Date
      2016-12-08
    • Related Report
      2016 Research-status Report
  • [Presentation] Malliavin calculus for conditional intensities of Hawkes processes2016

    • Author(s)
      竹内敦司
    • Organizer
      研究集会「確率解析とその周辺」
    • Place of Presentation
      九州大学 (福岡県福岡市)
    • Year and Date
      2016-11-09
    • Related Report
      2016 Research-status Report
  • [Presentation] Caloric morphism ― 熱方程式の解を保つ変換2016

    • Author(s)
      下村勝孝
    • Organizer
      第 59 回函数論シンポジウム
    • Place of Presentation
      静岡県男女共同参画センター あざれあ (静岡県静岡市)
    • Year and Date
      2016-10-08
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Polyharmonic Bergman spaces on half spaces2016

    • Author(s)
      西尾昌治, 下村勝孝
    • Organizer
      日本数学会
    • Place of Presentation
      関西大学千里山キャンパス (大阪府吹田市)
    • Year and Date
      2016-09-15
    • Related Report
      2016 Research-status Report
  • [Presentation] 半空間上の重み付き多重調和ベルグマン空間2016

    • Author(s)
      西尾昌治
    • Organizer
      2016年度ポテンシャル論研究集会
    • Place of Presentation
      大同大学 (愛知県名古屋市)
    • Year and Date
      2016-09-05
    • Related Report
      2016 Research-status Report
  • [Presentation] Lorentz不変形量に対する caloric morphism について2016

    • Author(s)
      下村勝孝
    • Organizer
      2016年度ポテンシャル論研究集会
    • Place of Presentation
      大同大学 (愛知県名古屋市)
    • Year and Date
      2016-09-05
    • Related Report
      2016 Research-status Report
  • [Presentation] Reproducing kernels for polyharmonic functions on half spaces2016

    • Author(s)
      Masaharu Nishio
    • Organizer
      International Conference on Complex Analysis and Related Topics, The 14th Romanian-Finnish Seminar
    • Place of Presentation
      Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania
    • Year and Date
      2016-06-20
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] L(a)-conjugates on parabolic Bloch spaces2016

    • Author(s)
      菱川洋介, 西尾昌治, 山田雅博
    • Organizer
      日本数学会年会
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] Toeplitz operators of radial measure symbol on the ball2016

    • Author(s)
      西尾昌治
    • Organizer
      研究集会「ポテンシャル論とその関連分野」
    • Place of Presentation
      北海道大学理学部(北海道札幌市)
    • Year and Date
      2016-01-31
    • Related Report
      2015 Research-status Report
  • [Presentation] 3次元以上の Lorentz 不変計量に対する caloric morphism について2016

    • Author(s)
      下村勝孝
    • Organizer
      研究集会「ポテンシャル論とその関連分野」
    • Place of Presentation
      北海道大学理学部(北海道札幌市)
    • Year and Date
      2016-01-31
    • Related Report
      2015 Research-status Report
  • [Presentation] Density of joint distributions for stochastic functional differential equations2015

    • Author(s)
      Atsushi Takeuchi
    • Organizer
      統計数理研究所共同研究集会「無限分解可能過程と関連する諸問題」
    • Place of Presentation
      統計数理研究所(東京都立川市)
    • Year and Date
      2015-12-03
    • Related Report
      2015 Research-status Report
  • [Presentation] 半空間上の多重調和関数に関する再生核について2015

    • Author(s)
      西尾昌治
    • Organizer
      RIMS 研究集会「再生核の応用についての総合的な研究」
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2015-10-07
    • Related Report
      2015 Research-status Report
  • [Presentation] Reproducing kernel for iterated parabolic operators on the upper half space with application to polyharmonic Bergman spaces2015

    • Author(s)
      西尾昌治
    • Organizer
      2015 年度ポテンシャル論研究集会
    • Place of Presentation
      名城大学名駅サテライト MSAT(愛知県名古屋市)
    • Year and Date
      2015-09-21
    • Related Report
      2015 Research-status Report
  • [Presentation] L(a)-conjugates on parabolic Bloch spaces2015

    • Author(s)
      菱川洋介, 西尾昌治, 山田雅博
    • Organizer
      2015 年度ポテンシャル論研究集会
    • Place of Presentation
      名城大学名駅サテライト MSAT(愛知県名古屋市)
    • Year and Date
      2015-09-21
    • Related Report
      2015 Research-status Report
  • [Presentation] Reproducing kernel for iterated parabolic operators on the upper half space2015

    • Author(s)
      西尾昌治
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      京都産業大学(京都府京都市)
    • Year and Date
      2015-09-13
    • Related Report
      2015 Research-status Report
  • [Presentation] Harmonic Bergman spaces with radial measure weight on the ball2015

    • Author(s)
      Masaharu Nishio and Kiyoki Tanaka
    • Organizer
      The Eighth Congress of Romanian Mathematicians
    • Place of Presentation
      Iasi, Romania
    • Year and Date
      2015-06-26
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research

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Published: 2015-04-16   Modified: 2020-03-30  

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