Some Fourier Multiplier Theorems

Since 1930's, the Fourier multiplier theory has been constantly a classical subject of analysis. The key part of the theory is to determine whether a function is a multiplier or not. The improvement of multiplier theorems can not only promote the development of harmonic analysis but also strengthen its applications to partial differential equations and other aspects of analysis.The main purpose of this dissertation is to give some Fourier multiplier theorems on H~p and L~p spaces. During the process of that, we mainly use methods and techniques of harmonic analysis, including interpolation of operators, analytic interpolation, dyadic decomposition. Particularly, the Torchinsky decomposition will play a key role in our proofs. Compared with past related works, the most important feature of the paper is to weaken the condition of the derivatives of the multipliers. In other words, the new theorems can be applied to more multipliers.This paper consists of two parts: the first part is concerned with H~p space cases and the second part is concerned with L~p space cases. In the first part, we firstly point out two kinds of description of H~p space and then use the Torchinsky decomposition to estimate some integrals of some functions. Using those estimations, we get the first main conclusion. In the second part, we firstly study L~1 case, and then combining this with the result of the first part, we get the second main conclusion by means of the analytic interpolation.