2004 Fiscal Year Final Research Report Summary
Harmonic Analysis for Orthogonal Expansions
| Project/Area Number |
15540161
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Kanazawa University |
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Principal Investigator |
KANJIN Yuichi Kanazawa University, Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (50091674)
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| Co-Investigator(Kenkyū-buntansha) |
TSUCHIYA Masaaki Kanazawa University, Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (50016101)
ICHINOSE Takasi Kanazawa University, Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (20024044)
SATO Shuichi Kanazawa University, Faculty of Education, Associate Professor, 教育学部, 助教授 (20162430)
TOHGE Kazuya Kanazawa University, Graduate School of Natural Science and Technology, Associate Professor, 自然科学研究科, 助教授 (30260558)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Hardy's inequality / real Hardy space / transplantation theorem / transplantation operator / Hankel transform / Cesaro operator / singular integral operator / fractional integral |
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| Research Abstract |
Our research results are summarized as follows. The head investigator Kanjin has obtained Hardy's inequality with respect to the Jacobi expansions. The classical Hardy's inequality is one of the most familiar inequalities on the Fourier coefficients of functions in the Hardy space on the unit disc. The inequality was originally proved by complex method. But, it is difficult to apply complex method to the study of orthogonal expansions. Our idea is to use the real Hardy space theory, especially the atomic decomposition characterization of the real Hardy space, and it allows us to discuss problems on inequalities with respect to orthogonal expansions. Our Hardy's inequality has proved by applying the atomic decomposition to the Jacobi function system. Further, he has studied the transplantation operators for the Hankel transform, and he obtained the boundedness of the operators on the real Hardy space.
The investigator Tsuchiya has investigated the diffusion equations with Ventcel'-Visik boundary conditions, and constructed their classical solutions. Ichinose has studied the spectral zeta function for the non-commutative harmonic oscillator and gotten the result that the zeta function has an analytic continuation to the whole complex plane as a meromorphic function. Sato has considered singular integrals with non-homogeneous kernels, and under certain additional conditions he obtained the weighted weak type (1,1) estimates for such singular integrals. Tohge has gotten some examples of holomorphic curves extremal to Cartan's defect relation.
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