Space Characterizations Of Singular Integral Operators Weighted Norm Inequality

The real variable theory of the Hp space is one of the most wealthiest fields of harmonic analysis from 1970's. The theory apply many kinds of maximal functions to characterize Hp space, this method is different from the complex method or the hamonic function method. The deeply developing stage about this theory is establishing the characterization structural theory of Hp space. The idea of the characterization structural theory is looking at the function spaces from microcosmic viewpoint, in other words regarding the element of Hp space as a overlap of a series of "basic element" According as the characterization structural theory we can make many problems about the hamonic analysis coming down to the simple instances.Many spaces's characterization structural theory have developed quite perfect, and part of the weighted spaces's characterization structural theory have fulfilled. In 1995, Lu Shanzhen and Yang Dachun had given the atom characterization of the weighted Herz type Hardy space, but so far there is not the molecole characterization of the weighted Herz type Hardy space. At first this paper solved the molecole characterization of the weighted Herz type Hardy space, as an application, author also gave the proof about the strong singular integral operator Tb's boundedness in weighted Herz type Hardy space.Along with the developing of the molecole characterization of the weighted Herz type Hardy space, the study of the singular integral operator get a plentiful harvest. But there is only a few fruits about the (t) type singular integral operator which had been braught forward by Yabuta in [37], and this operator has a profound partial differential coefficient background. The reason is that it is relatively complex. For the weighted case some operators have a little difficulty; For the case of the boundedness of this operator in Banach-value space, there is less study because the Banach-value space don't possess the good properties of the normal spaces. At last this paper recur to the Calderon-Zygmund characterization theory and the characterization theory of Hardy space, utilize the fine discussion, we get the 6(t) type singular integral operator's boundedness in the Banach-value weighted space LPB, (Rn) (1 p < ) and the Banach-value weighted space H1B, (Rn).At last, this paper use the molecole characterization of the weighted Hardy space to discuss the (t) type Calderon-Zygmund operator's boundedness in the weighted Hardy space, and proveing that (t) type Calder n-Zygmund operator is boundedness from Hp to Hp and from H1 to L1.