2001 Fiscal Year Final Research Report Summary
Multilateral research of stochastic analysis in infinite dimensional spaces
| Project/Area Number |
11440045
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
SHIGEKAWA Ichiro Graduate School of Science, Kyoto University, Professor, 大学院・理学(系)研究科(研究院), 教授 (00127234)
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| Co-Investigator(Kenkyū-buntansha) |
HINO Masanori Graduate School of Informations, Kyoto University, Lecturer, 大学院・情報学研究科, 講師 (40303888)
NOMURA Takaaki Graduate School of Science, Kyoto University, Associate Professor, 大学院・理学(系)研究科(研究院), 助教授 (30135511)
YOSHIDA Nobuo Graduate School of Science, Kyoto Univesity, Lecturer, 大学院・理学(系)研究科(研究院), 講師 (40240303)
AIDA Shigeki Graduate School of Engineering, Associarte Prefessor scinece, Osaka University, 大学院・基礎工学研究科, 助教授 (90222455)
UEKI Naomasa Graduate School of Human and Enviornmental Studies, Kyoto University, Associate Professor, 大学院・人間環境学研究科, 助教授 (80211069)
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| Project Period (FY) |
1999 – 2001
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| Keywords | semigroup domination / interturning property / square field operator / Littlewood-Paley ineguality / multiplier / second fundamental form / Hodge-Kodaira opoerator / logarithmic Sobolev ineguality |
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| Research Abstract |
We have accomplished the research of the semigroup domination and the intertwining property of semigroups. The semigroup domination stands for that |T^^→_tu|【less than or equal】T_t|u| holds for semigroup T^^→_t and T_t. Here u are supposed to be a vector valued function. Typical example is a differential form on a Riemannian manifold. The characterization in terms of the generator is known. We give here a sufficient condition in terms of square field operator. Crucial assumptions are the positivity and the locality. Intertwining property is the following property: for two generator L, L^^→, it holds that DL = L^^→D + R. We give necessary and sufficient conditions in terms of the semigroup and the resolvents. Our aim has been to reconstruct the.Bakry-Emery T_2 theory but we have succeeded in including diffusion processes with boundary condition and it gives a generalization of Bakry-Emery T_2 theory.
For application, we can make use of them to show the Littlewood-Paley inequality and the L^p multiplier theorem. In fact, we have considered the Brownian motion on an Riemannian manifold with boundary and show the Littlewood-Paley inequality for it under the assumption of positivity of the second fundamental form of the boundary. Making use of this, we can show the L^p boundedness of the Riesz transformation. L^p multiplier theorem is to give an sufficient condition on φ and A so that φ(A) is a bounded operator on L_p. Stein showed this for a generator of a symmetric diffusion process with φ being of Laplace transform type. We have shown that the same result holds for the Hodge-Kodaira operator on a Riemannian manifold.
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