Weight theory related to harmonic analysis on groups -in harmony with representation theory

与群调和分析相关的权重理论——与表示论一致

基本信息

批准号：
13640190
负责人：
KAWAZOE Takeshi
金额：
$ 2.18万
依托单位：
Keio University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

In this research we investigated the usage of fractional integrals in real analysis and also in harmonic analysis on semisimple Lie groups. E. Nakai, the investigator, studied boundedness of the fractional integral operators on various function spaces, especially, he generalized the operators and function spaces. T. Kawazoe, the head investigator, applied the theory of fractional integrals to construct real Hardy spaces on semisimple Lie groups. In this sense, we attained our aim of this research -in harmony with representation theory and real analysis.The theory of real Hardy spaces was generalized on the so-called space of homogeneous type satisfying the doubling condition, however, little works were done on the space of non-homogeneous type. In this research we took real rank one semisimple Lie groups G as an example of the space of non-homogeneous type, and introduced real Hardy spaces for K-bi-invariant functions. This corresponds to harmonic analysis on Fourier-Jaccobi transforms … More and on the weight theory with exponential growth order. We first, by using the Abel transform, reduced the Fourier-Jaccobi analysis to Euclidean one on the real line R. Since the Abel transform was written by fractional integrals, the method of real analysis related to fractional integrals were useful in this analysis. Next, five extended the theory of fractional integrals and derivatives and then characterized real Hardy spaces defined by maximal functions and atoms. Our main theorem is the following. Let W be the modified Abel transform with Exp(ρx)-multiplication and V the inverse of W. Then, we see H1 (G//K) ⊂V (H1(R)), where H1(G//K) and HI(R) are respectively the real Hardy spaces on G and R defined by the radial maximal functions. Moreover, let A1(G//K)+ denote the atomic Hardy space on G defined by restricting the moment condition of atoms only for ones with redius <1. Then it follows that H1(G//K)= V(H1(R))∩A1(G//K)+. In the proof we applied some methods in real analysis related to fractional integrals and atomic decompositions. Less

在这项研究中，我们调查了分数积分在实际分析中的用法以及对半圣母谎言组的谐波分析。研究人员E. Nakai研究了分数积分运算符在各种功能空间上的界限，尤其是他概括了操作员和功能空间。首席调查员T. Kawazoe应用了分数积分理论，以在半岛谎言组上构建真实的耐寒空间。从这个意义上讲，我们达到了这项研究的目的 - 与代表理论和真实分析和谐相处。真实耐寒空间的理论是在满足双倍条件的所谓均质类型空间上概括的，但是，对非殖民类型的空间几乎没有完成。在这项研究中，我们以实际等级的半圣谎言组G作为非均匀类型的空间的一个例子，并引入了K-Bi-Invariant功能的真实耐寒空间。这对应于对傅里叶 - 贾科比变换的谐波分析……更多，以及具有指数增长顺序的重量理论。首先，通过使用ABEL变换，将傅立叶 - jaccobi分析降低到了实际线路上的欧几里得。由于亚伯变换是由分数积分编写的，因此与分数积分相关的真实分析方法在此分析中很有用。接下来，五个扩展了分数积分和衍生物的理论，然后表征了由最大函数和原子定义的真实耐寒空间。我们的主要定理是以下内容。令W为exp（ρx） - 多授课和V的修改ABEL变换。然后，我们看到H1（g // k）⊂V（h1（r）），其中H1（g // k）和HI（r）分别是G和h（r）在G和R上由radial Maximal函数定义的真实硬度空间。此外，让A1（g // k）+表示G在G上的原子硬度空间，通过仅限制了原子的矩条，仅适用于redius <1的原子。然后得出H1（g // k）= v（h1（r））∩a1（g // k）+。在证明中，我们在与分数积分和原子分解有关的实际分析中应用了一些方法。较少的