2007 Fiscal Year Final Research Report Summary
Analysis of Banach function spaces in view point of martingale theory
| Project/Area Number |
17540152
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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 | University of Toyama |
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Principal Investigator |
KIKUCHI Masato University of Toyama, Graduate school of Science and Engineering, Associate Professor (20204836)
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| Co-Investigator(Kenkyū-buntansha) |
KUBO Fumio University of Toyama, Graduate school of Scienceand Engineering, Professor (90101188)
IZUMISAWA Masataka Tokai University, Faculty of Science, Professor (50108445)
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| Project Period (FY) |
2005 – 2007
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| Keywords | martingale / Banach function space / rearrangement-invariant space / norm inequality / Boyd index |
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| Research Abstract |
The research results of our project are as follows: In what follows, we let(X,‖・‖x) denote a Banach function space over a probability space
・ We introduced the Banach space K(X) of martingales that extends K_P(2≦p<∞) introduced by A.M. Garsia, and studied some norm inequalities for martingales f=(f_n) involving the norms of f in K(X) and X.
・ Given a uniformly integrable martingale f=(f_n), letAf(Af_n)denote the martingale induced by |f∞|, where f∞=lim_n f_n. We established a characterization of those Banach function spaces X such that f∈K(X) ⇔ Af∈K(X).
We extended the Burkholder-Davis-Gundy inequality, and established a characterization of those Banach function spaces X for which the inequality‖Mf‖x≦Cx‖Sf‖x holds for any martingale f, where Mf denotes the maximal function of f and Sf denotes the square function of f.
・ Let f =(f_n) be a martingale and let Φ:R→[0,∞) be a convex function such that Φ(0) =0 and Φ(t)=Φ(-t) for all t≧0. Because Φ(f) =(Φ(f_n)) is a submartingale, it can be decomposed into a sum of a martingale g=(gn) and a predictable increasing process h=(h_n) such that h_0=0 (Doob decomposition). We established a characterization of those Banach function spaces X for which the inequalities‖h_∞‖x≦C_x,_Φ‖Mf‖x and sup_n‖g_n‖x≦C_x,_Φ‖Mf‖x hold.
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Research Products
(8 results)
