1998 Fiscal Year Final Research Report Summary
Harmonic Analysis and its Applications
| Project/Area Number |
08454022
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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 |
解析学
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| Research Institution | Tohoku University |
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Principal Investigator |
IGARI Satoru Tohoku University, The Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50004289)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Kazuyuki Tohoku University, The Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (60004397)
TAKAGI Izumi Tohoku University, The Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40154744)
MASUDA Kyuya Tohoku University, The Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10090523)
TACIZAWA Kazuya Tohoku University, The Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (80227090)
ARAI Hitoshi Tohoku University, The Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10175953)
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| Project Period (FY) |
1996 – 1998
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| Keywords | maximal function / wavelet / degenerate elliptic operator / minimal surface / C^*-algebra / Fourier transform / Navier-Stokes equation |
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| Research Abstract |
The problem of the harmonic analysis is often reduced to the invitational of translation invariant operators. As a useful real analysis method of such operators, there is the singular integral theory. However, we pay attention to the important translation invariant operators which aren't satisfied with the category of that theory. There, the Kakeya maximal function plays an important role an auxiliary function. Igari defines a maximal function which has a special base, and gave an estimate which gives the best possible estimate for radial functions, and also a partially Bourgain's result.
Arai studied, form the viewpoint of the harmony analysis, the elliptic partial differential operators which degenerate in all points in the boundary, such as operators in pseudo-convex domain of the Stein manifolds and in the manifolds with theta-structure. He gets a basic result in the harmony analysis of such operators and is preparing a paper on it. Also, he found he fact which has a possibility to solve the pointwise Fatou problem for functions of several complex variables.
It is studied for a long time the uniqueness of the solution of the Navier-Strokes equation, but it is still an unsolved major theme. Masuda introduced a quite new way which is of real variable method for this problem, and found a beginning of solving the problem. This result was reported in the international conference held in Verona. Also, he solved the conjecture on the classification of the minimal surfaces of constant curvature in two dimensional complex space from by the collaboration with K.Kenmotsu..
Saito developed an important tool for the study of the monotone complete C*-algebra by giving an elementary proof for a regularity of the Rickart C*-algebra. He also gets a result about the completely additive measures on von Neumann algebras.
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