Potential theory for parabolic equations with functional analysis

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540220
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka 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): SAKAN Ken-ichi 大阪市立大学, 大学院理学研究科, 准教授 (70110856) ; TAKEUCHI Atsushi 大阪市立大学, 大学院理学研究科, 准教授 (30336755) ; SHIMOMURA Katsunori 茨城大学, 理学部, 教授 (00201559)
• Co-Investigator(Renkei-kenkyūsha): SUZUKI Noriaki 名城大学, 理工学部, 教授 (50154563) ; YAMADA Masahiro 岐阜大学, 教育学部, 准教授 (00263666) ; MASAOKA Hiroaki 京都産業大学, 理学部, 教授 (30219315)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: ポテンシャル論 / 熱方程式 / 分数ベキ作用素 / ベルグマン空間 / テープリッツ作用素 / ブロッホ空間 / シャッテン族作用素 / 調和双対

Outline of Final Research Achievements

In this research, we studied parabolic operators, including the heat equation and the fractional Laplacian, by using potential theoretic and functional analytic methods. The mean value properties obtained by potential theory play important roles, which enables us to discuss spaces of solutions of parabolic equations in the functional analytic way. In fact, we showed that they admit the symmetric reproducing kernels, which are called parabolic Bergman kernels. As main results of this research, we characterized for the Toeplitz operator to be bounded, compact, snd belonging to the Schatten classes by using some scaling invariant property of parabolic type. We also discussed the various notions of harmonic conjugates of parabolic type and clarified their relations.

Free Research Field

ポテンシャル論