2019 Fiscal Year Final Research Report
The weight theory under some recent innovations
| Project/Area Number |
15K04918
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tsukuba University of Technology (2016-2019)
The University of Tokyo (2015) |
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Principal Investigator |
Tanaka Hitoshi 筑波技術大学, 障害者高等教育研究支援センター, 講師 (70422392)
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Keywords | fractional integrals / maximal operator / Morrey space / n-linear embedding / weight theory |
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| Outline of Final Research Achievements |
In 1995, Nazarov, Treil and Volberg established the bilinear embedding theorem which is simple and quite interesting.In this research, we expect to consider the multi-linear version of this theorem.In 2015, using the single Wolff potential, we verified the trilinear embedding theorem.In 2016, applying the Wolff potential repeatedly, we succeeded to obtain multi-Wolff potential and proved n-linear embedding theorem.In 2019, using Yabuta's lemma, which can be proved by a clever iteration argument, we showed n-linear embedding theorem with respect to the dyadic rectangles (not the dyadic cubes). This concerns with strong maximal operators.
Moreover, in 2018, we characterized weighted Morrey norm inequality for fractional integral operators with the power weights.
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| Free Research Field |
解析学基礎
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| Academic Significance and Societal Importance of the Research Achievements |
作用素の評価を荷重(weight)付きというより一般化された設定の下で研究することは，基本的であり応用上も重要視されています．それは，作用素の値域に関する情報を陽的に与えるものであると解釈できて，作用素のより深い理解に繋がるものです．
本研究では，いくつかの作用素について，荷重付ノルム不等式が成立するために荷重が満たすべき条件を明らかにし，その特徴づけを与えました．特に，n重線形埋蔵低利の証明はこの方向の研究の最後の部分にあたり，一つの完成形を与えました．
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