A study on wavelets and function spaces with variable exponent based on real analysis

2022 Fiscal Year Final Research Report

• Project/Area Number: 15K04928
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokyo City University (2019-2022); Okayama University (2015-2018)
• Principal Investigator: Izuki Mitsuo 東京都市大学, 共通教育部, 准教授 (80507179)
• Project Period (FY): 2015-04-01 – 2023-03-31
• Keywords: ウェーブレット / 変動指数 / Muckenhouptの理論 / BMO

Outline of Final Research Achievements

One of the important result is a study on Herz spaces with variable exponents joint work with Professor Takahiro Noi. We have constructed a method which shows boundedness of some fundamental operators on function spaces involving variable exponent and Muckenhoupt weight. In addition, we have defined a new Hardy Space associated to critical Herz spaces with variable exponent and proved some basic properties of the space. Additionally we have defined two weighted Herz spaces with variable exponents and given equivalent norms. As a result, we have constructed a basic analysis method on those spaces.
One of the another result is a joint work with Professor Yoshihiro Sawano, Doctor Toru Nogayama and Professor Takahiro Noi, which have obtained several results on local Muckenhoupt weights with variable exponent and corresponding function spaces.

Free Research Field

実解析

Academic Significance and Societal Importance of the Research Achievements

今後、より多くの自然現象の数学モデル化や工学における諸分野におけるウェーブレット理論や変動指数解析の応用を進めていく上で、その枠組みとなる多用な関数空間が求められることになります。Herz空間において、変動指数とMuckenhoupt荷重の理論を用いて変動指数型の荷重つきの新たな関数のクラスを複数定義してそれらの性質を明らかにできたこと、実解析における最新の理論ともいえる局所変動指数型Muckenhouptの荷重理論の研究に取り組み、その荷重と変動指数の両方をもつ幾つかの関数空間において一定の成果を得られたことは大変意義があります。