Studies on Hardy spaces by real analytic methods

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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.