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Studies on Hardy spaces by real analytic methods

Hardy空间的实解析方法研究

基本信息

批准号：
09640146
负责人：
KANEKO Makoto
金额：
$ 0.9万
依托单位：
TOHOKU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

The Hardy spaces constructed in an n-dimensional Euclidean space have been studied by many authors since a long time ago as well as those built on n-dimensional torus. In this research we have pointed out that the both of two have very similar constructions through the investigation of the transference problem.When a bounded function defined on an n-dimensional Euclidean space is given, we have an operator called Fourier multiplier operator which is defined by multiplying the Fourier transform of an object function or a distribution by the bounded function. On the other hand, if we consider the restriction of the given bounded function to the n-dimensional lattice, then we have a Fourier multiplier operator in the frame of Fourier series arguments.A countable number of bounded functions make a sequence of Fourier multiplier operators in both frames of Fourier transform and Fourier series. Each of them constructs the associated maximal operator. We have succeeded to prove that the continuity of the maximal operator in the frame of Fourier transform argument from a Hardy space to a weak Lebesgue space implies the continuity of the counterpart maximal operator in the setting of Fourier series argument.Furthermore, we have studied the maximal operator obtained by the family of convoluted functions by an integrable function of a given sequence of bounded functions. We have gained an simple proof of showing that the continuity of the above maximal operator is reduced from the continuity of the maximal operator defined by the sequence of initially given and nonconvoluted bounded functions.

构造在n维欧氏空间上的哈代空间和构造在n维环面上的Hardy空间，一直以来都是许多学者研究的对象。本文通过对迁移问题的研究，指出两者具有非常相似的结构：当给定一个定义在n维欧氏空间上的有界函数时，我们有一个称为Fourier乘子算子的算子，它是通过将目标函数或分布的Fourier变换与有界函数相乘而定义的。另一方面，如果考虑给定的有界函数对n维格的限制，则在Fourier级数变元框架下存在Fourier乘子算子，可数个有界函数构成Fourier变换框架和Fourier级数框架下的Fourier乘子算子序列.它们中的每一个构造了相关的极大算子。我们成功地证明了极大算子在从哈代空间到弱Lebesgue空间的Fourier变换变元框架下的连续性蕴涵了相应的极大算子在Fourier级数变元框架下的连续性，并进一步研究了由给定有界函数列的可积函数构成的卷积函数族所得到的极大算子.我们得到了一个简单的证明，证明了上述极大算子的连续性是由初始给定的非卷积有界函数序列所定义的极大算子的连续性退化而来的.