Boundedness Of Singular Integral Operators On Several Multi-Parameter Function Spaces

The theory of singular integral operators plays an important role in harmonic analysis.So far,the study of the boundedness of Calderón-Zygmund singular integral operator on various single-parameter function spaces has been fruitful.However,the boundedness in multi-parameter functions spaces needs to be further investigated.This paper is divided into four chapters.In Chapter 1,we introduce the background of the theory of spaces of homogeneous type,the theory of singular integral operators and the theory of multi-parameter reconciliation analysis,as well as the current status of the study of boundedness of CalderónZygmund singular integral operators on different function spaces.On this basis,the main problem to be solved in this paper is presented.In Chapter 2,a necessary and sufficient condition for the boundedness of the generalized product Calderon-Zygmund operator on product Lipschitz function spaces T₁1=T₂1=0 is established over product RD spaces,where the quasi-metric satisfies the H?lder regularity condition and the measure satisfies the double condition and the reverse doubling condition.The key idea is to develop the Littlewood-Paley theory of the product Lipschitz function space,which includes the inscription of the special product Besov space and the density argument in the weak sense.In Chapter 3,we establish the product T1 theorem on product homogeneous Besov spaces（?） and product homogeneous Triebel-Lizorkin spaces（?） with only half of the usual smoothnessover Euclid spaces Rⁿ×R^m.The key idea is to obtain the Plancherel-Polya characterization of the product Besov and Triebel-Lizorkin spaces by the discrete Calderon type identity and the almost orthogonality estimate.Moreover,the index p and q includes the case that less than or equal to 1.In Chapter 4,we conclude the paper with outlooks.