Two Types Of Singular Integral Operators Commutators Boundedness,

In this thesis, we discuss the boundedness of two types of commutators on weighted Herz type spaces, which are [b, T] generated by generalized Calderon-Zygmund operators T with Lipschitz function b and a class of Marcinkiewicz integral commutatorsμ_Ω,b generated by BMO(Rⁿ) functions and Marcinkiewicz integrals with rough kernels. This thesis consists of the following parts.In the first chapter, using the decomposition of Herz type Hardy spaces in terms of central atoms and the properties of A₁, the boundedness of the commutator [b,T] generated by generalized Calderon-Zygmund operators T with Lipschitz function b is studied. We show that [b,T] is bounded from homogeneous Herz type Hardy space HK_q1^α,p₁(ω₁,ω₂^q₁) to homogeneous Herz space K_q2^α,p₂(ω₁,ω₂^q₂). And we also obtain that [b,T] is bounded from homogeneous Herz type Hardy space to homogeneous Herz space on critical points. For the nonhomogeneous Herz spaces, we have the similar results.In the second chapter, the boundedness on homogeneous weighted Herz spaces is established for a class of Marcinkiewicz integral commutatorsμ_Ω,b generated by BMO(Rⁿ) functions and Marcinkiewicz integrals with rough kernels . The main results are deduced from the definitions of K_q2^α,p₂(ω₁,ω₂^q₂). By the Minkowski integral inequality and the Jensen inequality to control some inequalities, we obtain the boundedness of Marcinkiewicz integral commutatorsμ_Ω,b on weighted Herz spaces when a, q satisfy two kinds of different conditions.