With Distortions The Convolution Hardy Space Of The Atomic Decomposition

In the development process of Harmonic analysis. Many important operators (like Hilbert transformation) is not bounded from L1 to L1,but it is bounded from H1 to L1. So we can discuss the characters of the operrators from researching the characters of H sometimes. The classic Hardy space is function apace which satisfy some conditions in the unit ciecle or the upper half plane. Because the real part and the imaginary part of the analytical function are conjugate harmonic functions,then their boundary function is conjugate functions. This is the simplest and most typical singular integral operator. So The classic Hardy space is associated with the Harmonic analysis. The classic Hardy space theory is very concise because of the tool of analytical function,but it is hard to extend to the higher-dimention. Since the real method of Harmonic analysis come out,the classic Hardy space theory can extend to the higher-dimention. The characters of the classic HP space (like atomic decomposition,molecule decomposition,dual space and so on) have been given in some papers. In this paper we research the atomic decomposition of HP(Cn) space associated with twisted convolution.Firstly, We introduce the meaning and status of researching the Hp space. we also introduce some prepare knowledge and the content and the structure arrangement of this paper. Secondly, We give the definition of the twist convolution. Then we define the Grand maximal function under the meaning of twist convolution. After that we use the boundedhess of the Grand maximal function to character the HP(C") space associated with twisted convolution.Thirdly, We proof the existence of the atomic decomposition of Hp(Cn) space associated with twisted convolution.Nestly, We use the existence of the atomic decomposition of HP(C") space associated with twisted convolution to research the boundedness of Weyl multiplier.Lastly, we sum up this paper and give some views of the following-up research.