Characterization Of Hardy Space Poisson Maximal Function Associated With Twisted Convolution

Classical Hardy space has an important significance in proving the boundedness of singular integral operators. A significant progress of Harmonic analysis is the characterizations of Hardy spaces by the real variable methods that completely out of complex analysis method. These characterizations including maximal function characterization, square function characterization, Riesz transform characterization, atomic characterization and molecular characterization and so on.Study on the Hardy space has a long history, the classical Hardy space is in the unit circle or on the upper half plane defined by the complex analysis method. The analysis of these theories in the study of the classical Fourier space problem plays an important role. First of all, this work is e.m.stein and G. Weiss. They established in the early 1960s n-dimensional Hardy space theory is not method based on complex analysis, but the method of harmonic functions. Regardless of the classical Hardy space is the n-dimensional Hardy space theory, until the 1970s is a breakthrough in the development, making in recent decades research on Hardy space become study on harmonic analysis in the booming field of, one of the reasons lies in consolidation method has now entered the theory study of Hardy space. The first attempt is D.L.Burk-holder, R.F.Gundy and M.L.Silverstein, they used the method of probability theory gives a one-dimensional Hardy space consolidation characteristics. Then, and e.m.stein C.Fefferman put the results in real analysis method is generalized to the n-dimensional, and that can be maximal functions with complex analysis and harmonic function method independent of various forms to portray the Hardy space, marking the establishment of the theory of consolidation of the Hardy space.Similar as Fourier analysis, the fundamental element in Hardy space is atom (satisfies size condition, compact support condition, vanishing moment condition). Atomic decomposition plays an important role in the Hardy spaces, it can simply many complicated problems, including Poisson maximal characterization. Poisson maximal characterization has many important applications in the classical Hardy space. Is it possible to get the Poisson maximal characterization of Hardy space associated with twisted convolution? The Poisson kernel of twisted convolution is very different from the Poisson kernel of Laplacian, we must find new methods to get our result.This paper will study the characterization of Hardy spaces associated with twisted convolution by Poisson maximal function. We also define area integral and Littlewood-Paley g-function by the Poisson semigroup generated by twisted Laplacian, then the characterizations of Hardy spaces associated with twisted convolution by area integral and Littlewood-Paley g-function will be given through Poisson maximal function.