Studies on Hardy spaces by real analytic methods

构造在n维欧氏空间上的Hardy空间以及建立在n维环面上的Hardy空间早已被许多作者所研究。在这项研究中，我们通过对迁移问题的研究，指出两者具有非常相似的结构，当定义在n维欧氏空间上的有界函数时，我们有一个称为傅里叶乘子算子的算子，它是由目标函数或分布的傅立叶变换乘以有界函数定义的。另一方面，如果考虑给定有界函数在n维格上的限制，则在傅里叶级数的框架下有一个傅里叶乘子算子，在傅里叶变换和傅里叶级数的两个框架下，可数个有界函数构成一个傅里叶乘子算子序列。它们中的每一个都构成了相关的极大算子。我们成功地证明了从Hardy空间到弱Lebesgu型空间的傅里叶变换框架中的极大算子的连续性蕴含着相应的极大算子在傅里叶级数的背景下的连续性，并进一步研究了由给定的有界函数序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降低的。

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.