2004 Fiscal Year Final Research Report Summary
Analysis of Banach function spaces by means of martingale method
| Project/Area Number |
14540164
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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 | National University Corporation Toyama University |
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Principal Investigator |
KIKUCHI Masato Toyama University, Faculty of Science, Associate Professor, 理学部, 助教授 (20204836)
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| Co-Investigator(Kenkyū-buntansha) |
KOBAYASHI Kusuo Toyama University, Faculty of Science, Professor, 理学部, 教授 (70033925)
KUBO Fumio Toyama University, Faculty of Science, Professor, 理学部, 教授 (90101188)
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| Project Period (FY) |
2002 – 2004
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| Keywords | martingale / Banach function space / rerrangement-invariant function space / norm inequality / Boyd index |
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| Research Abstract |
The research results of our project are as follows :
・Let X be a Banach function space over a non-atomic probability space. Given a martingale f=(f_n), denote by Sf the square-function of f. I gave a necessary and sufficient condition on X for the Burkholder square-function inequality c‖f_∞‖_X【less than or equal】‖Sf‖_X【less than or equal】C‖f_∞‖_X to hold, where f_∞ denotes the almost sure limit of f.
・Given a uniformly integrable martingale f=(f_n), let Af=(Af_n) denote the martingale generated by the absolute value of f_∞. I gave a necessary and sufficient condition on a Banach function space X for that Sf and S(Af) belong to X simultaneously.
・Let X be a rearrangement-invariant Banach function space over a non-atomic probability space. I established various new martingale inequalities in the rearrangement-invariant function spaces X, H_p(X) and K(X), where H_p(X) and K(X) are defined so as to have a deep and suitable relation with X.
・Let X be a Banach function space over a non-atomic probability space. I gave a necessary and sufficient condition on X for the Davis inequality ‖Mf‖_X【less than or equal】C‖Sf‖_X to hold, where Mf denote the maximal function of f. As a result, it follows that if this inequality holds, then the reversed inequality ‖Mf‖_X【less than or equal】C‖Sf‖_X also holds.
・Give a martingale f=(f_n), define a process θf=(θf_n) by setting Of_n=sup_<0【less than or equal】n【less than or equal】m<∞>E[|f_m-f_<n-1>‖F_n]. I gave necessary and sufficient conditions for some inequalities involving the norm of θf in a Banach function space X.
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