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.

从很久以前，许多作者以及建立在n维圆环的北方欧几里得空间中已经研究了在n维欧几里德空间中建造的强壮空间。在这项研究中，我们指出，通过研究转移问题，两个中的两个都具有非常相似的结构。当给出了n维欧几里得空间上定义的有界函数时，我们拥有一个称为傅立叶乘数运算符的操作员，该运算符通过将对象函数的傅立叶变换或通过边界函数乘以分布的傅立叶变换来定义。另一方面，如果我们考虑将给定有界函数限制到n维晶格中，那么在傅立叶串联参数框架中，我们有一个傅立叶乘数运算符。一个可数的有界函数使傅立叶型乘数运算符序列在傅立叶变换框架和傅立叶系列中。他们每个人都构建了相关的最大操作员。我们成功地证明了最大运算符的连续性在傅立叶变换的框架中，从强硬的空间到弱的lebesgue空间，这意味着在傅立叶级数参数的设置中，对应物最大运算符的连续性。Furthermore的设置。我们已经研究了由乘积功能的最大运算符获得的最大运算符。我们获得了一个简单的证明，证明上述最大运算符的连续性是从最初给定和未卷入的有界函数的序列定义的最大运算符的连续性降低的。