Theory and applications of BMO and related function spaces on spaces of homogeneous type

齐次型空间上的BMO及相关函数空间的理论与应用

• 批准号: 11640165
• 负责人: NAKAI Eiichi
• 金额: $ 1.6万
• 依托单位: Osaka Kyoiku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640165/
• 关键词: bounded mean oscillation; space of homogeneous type; pointwise multiplier; fractional integral; Riesz potential; Orlicz space; Campanato space; Morrey space

1. The theory of pointwise multipliers on the function space Lp is well known. In this research, we have developed this theory on several function spaces ; Lorentz, Orlicz, Morrey, BMO, and Campanato spaces. The pointwise multiplier is simple and basic. So we can examine properties of spaces of homogeneous type by studying it.2. The boundedness of singular integral operators and the Riesz potential are useful for the theory of partial differential equations and are studied by many authors. In particular, the boundedness of the Riesz potential (fractional integral) from L^p to L^q is well known as the Hardy-Littlewood-Sobolev theorem. In this research, we introduce generalized fractional integrals and extend this boundedness on several function spaces ; Orlicz, BMO_φ, Campanato, Morrey, weak-Orlicz and generalized Hardy spaces.3. We have developed the method to redefine the quasi-distance of the space of homogeneous type on which the function space is not change, and we have the following as applications :(1) We have other results for the theory of pointwise multipliers on Campanato spaces.(2) On several operators used in the theory of partial differential equations, for example Kohn Laplacian, some results on the normal space of homogeneous type are adaptable to general spaces of homogeneous type.(3) On the fractional integral and derivative, some results on the normal space of homogeneous type are adaptable to general spaces of homogeneous type.

1.函数空间Lp上的逐点乘子理论是众所周知的。在这项研究中，我们已经开发了这个理论的几个功能空间;洛伦兹，Orlicz，Morrey，BMO和Campanato空间。逐点乘子是简单而基本的。因此，我们可以通过研究齐型空间来考察齐型空间的性质.奇异积分算子的有界性和Riesz位势在偏微分方程理论中有重要的应用，许多作者都对它们进行了研究。特别地，Riesz势（分数次积分）从L^p到L^q的有界性被称为Hardy-Littlewood-Sobolev定理。在本研究中，我们引入了广义分数次积分，并将其推广到Orlicz，BMO_φ，Campanato，Morrey，weak-Orlicz和广义哈代空间.本文给出了函数空间不变的齐型空间的拟距离的定义方法，并得到如下应用：（1）给出了Campanato空间上点态乘子理论的其它结果。(2)关于偏微分方程理论中的几种算子，如Kohn-Laplacian算子，齐型正规空间上的一些结果适用于一般齐型空间。(3)关于分数次积分和导数，齐型正规空间上的一些结果适用于一般齐型空间。

项目成果

Chikako Harada and Eiichi Nakai: "The square partial sums of the Fourier transform of radial functions in three dimensions"Scientiae Mathematicae Japonicae Online. Vol.5 (to appear). 329-339 (2001)

Chikako Harada 和 Eiichi Nakai：“三维径向函数的傅立叶变换的平方部分和”Scientiae Mathematicae Japonicae Online。

Eiichi Nakai : "A characterization of pointwise multipliers on the Morrey spaces"Scientiae Mathematicae. 3. 445-454 (2000)

Eiichi Nakai：“莫雷空间上的点乘子的表征”数学科学。

Eiichi Nakai: "On generalized fractional integrals"Taiwanese Journal of Mathematics. 5. 587-602 (2001)

中井英一：《论广义分数积分》台湾数学杂志。

Eiichi Nakai: "On generalized fractional integrals on the weak Orlicz spaces, BMO_φ, the Morrey spaces and the Campanato spaces"Proceedings of the Conference on Function Spaces, Interpolation Theory, and related topics, Lund University, Sweden. (印刷中).

Eiichi Nakai：“关于弱 Orlicz 空间、BMO_φ、Morrey 空间和 Campanato 空间的广义分数积分”，函数空间、插值理论和相关主题会议论文集，瑞典隆德大学（正在出版）。

Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type"Scientiae Mathematicae Japonicae. Vol.54 (Scientiae Mathematicae Japonicae Online, Vol.4 (2001), 901-915). 473-487 (2001)

Eiichi Nakai：“关于齐次类型空间上 Orlicz 空间中的广义分数积分”Scientiae Mathematicae Japonicae。