Generalized Marcinkiewicz Integral Operator Bounded

This dissertation mainly consists of two parts.In Chapter one ,we study the Marcinkiewicz integral operatorWhen satisfying certain cancellation condi-tions.We prove that,for 1 < p < ∞,μ_Ω,α is bounded from the homogeneous Sobolev space L^p_a to the Lebesgue space L^p, and, for 2 ≤ p < ∞,Marcinkiewicz integral operators μ_Ω,λ,α and μ_Ω,S,α which are related to the function and the area integral S respectively are bounded operators from the homogeneous Sobolev space L^p_a to the Lebesgue space L_p In Chapter two, we prove for the corresponding Marcinkie-wicz integral operator on product spaces is bounded from...