Anisotropic Weighted Hardy Space Characterizations

In this thesis, we introduce the anisotropic weighted Hardy space associated with the very general discrete group of dilations. The atomic and molecular decompositions of this spaces are disscused. And the boundedness of some singular integral operators is obtained for application. The conclusions in this thesis extend the theory of anisotropic Hardy spaces. We put this paper into the following three parts:In the first part, the anisotropic weighted (p, q, s,ω) atoms are introduced. On the basis of the atoms, the atomic decomposition of the anisotropic weighted Hardy space is established.In the second part, the anisotropic weighted (p, q,0,ε,ω) moleculars are introduced. Then by the properties of the weighted function, the molecular decomposition of the anisotropic weighted Hardy space is obtained.In the last part, for application, the boundedness of generalized Calderon-Zygmund singular integral operators which satisfies certain conditions on the anisotropic weighted Hardy spaces is proved.