On interpolation and extrapolation theorem in Orlicz spaces
| Project/Area Number |
09640206
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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 |
解析学
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| Research Institution | Oita University |
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Principal Investigator |
HIRO-O Kita Fuculty of Education Professor, 教育学部, 教授 (20224941)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEMOTO Yoshio Nihon Bunri Univ.Fuculty of Engineering Professor, 工学部, 教授 (20140965)
KEMOTO Nobuyuki Oita University Fuculty of Education Asociated Professor, 教育学部, 助教授 (70161825)
MORI Naganori Oita University Fuculty of Education Professor, 教育学部, 教授 (40040737)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | Fourier series / Orlicz space / a.e.convergence / maximal function / オ-リッツ空間 / フーリェ級数 |
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| Research Abstract |
The first research product of our study in 1998 is as follows. Let *(t) be an increasing function defined on the interval [O,*) satisfying *(0) 0. Let L*(T) be an Orlicz space on T = [-"p, pi}. We denote by S_n(f, x) the n-th partial sum of the Fourier series of an integrable function f. And S^*(f) means the Fourier maximal operator. When *(t) exp(V^<gamma>) - 1, (-gamma > 0), we denote by L(expf^<gamma>) the Orlicz space generated by this function *(t). It was proved that if f is a function in the Orlicz space L(expt^<gamma>), then S^*.(f) is in L(expt^<gamma>/^(gamma^<c+>^1^)). This result was already shown in our paper in detail (Acta Math. Hungar. 1994).
A generalization of the result mentioned above can be considered. In our previous paper, Young function * was restricted. However in our recent paper this restriction was removed. Our main idea of the proof of this result is an interpolation theory of quasi linear operators in Lorentz spaces. Let *(t) be a rapid]y increasing Young function and L*(T) be an Orlicz space defined by tbis *(t). We could find the sharp Young function * such that S^*(f) is in L*(T) for all f in L*(T).
The second research product of our study is a result of almost everywhere convergence of Fourier series of functions in an Orhicz space near to Zygmund class L log L.We have the following result. If
l^t_1(*(u))/du<greater than or equal>a_0 log(1+jogt) for t>1
holds, then we get ||*(S^*(f)||LA^1 for all f * L* (see H.Rita [Kil]).
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Report
(3 results)
Research Products
(12 results)
