2006 Fiscal Year Final Research Report Summary
A characterization of weighted Sobolev spaces and its applications
| Project/Area Number |
16540133
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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 | Hokkaido University |
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Principal Investigator |
TACHIZAWA Kazuya Hokkaido Univ. Fac. of Sci., Fac. of Sci., Asso. Prof., 大学院理学研究員, 助教授 (80227090)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAZI Takahiko Hokkaido Univ., Fac. of Sci., Prof., 大学院理学研究員, 教授 (30002174)
HAYASHI Mikihiro Hokkaido Univ., Fac. of Sci., Prof., 大学院理学研究員, 教授 (40007828)
OZAWA Tohru Hokkaido Univ., Fac. of Sci., Prof., 大学院理学研究員, 教授 (70204196)
HORIUCHI Toshio Ibaraki Univ., Fac. of Sci., Prof., 理学部, 教授 (80157057)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Sobolev space / wavelet / weighted space / Triebel-Lizorkin space |
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| Research Abstract |
1.We proved a weighted version of the Sobolev-Lieb-Thirring inequality. As an application we showed a weighted LP-Sobolev-Lieb-Thirring inequality which is a new result even in the non-weighted case.
2.We proved that the wavelet basis is a greedy basis in weighted Triebel-Lizorkin spaces. As an application we determined the approximation spaces of the weighted Triebel-Lizorkin spaces by means of the non-linear approximation by wavelets.
3.We proved a wavelet characterization of weighted Herz spaces. Tang and Yang showed a vector valued inequality in weighted Herz space although there are some mistakes in their condition on weights. We gave a correct condition on weights and proved the wavelet characterization.
4.We studied the decay estimate of the solution of a non-linear elliptic partial differential equations with a potential. We characterize the class of weights in the decay estimate of the solution by the potential. We proved the global existence of a solution with small amplitude for several dispersive equations such as modified Boussinesq equations, improved Boussinesq equations and semi relativistic Hartree equations.
5.We studied the existence of a singular solution and its property of a semilinear degenerate elliptic equation with p-harmonic operator in the principal term. In particular we investigated a linearized degenerate elliptic operator and proved the non-negativeness of the smallest eigenvalue and its relation to the Hardy type inequality.
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