The Properties Of Marcinkiewicz Integral Operators And Commutators

In this dissertation,the properties of Marcinkiewicz integral oper-ators and commutators are considered.In the first chapter,we prove that the Marcinkiewicz integral op-eratorμΩis an operator of type(H^p,∞,L^p,∞)for(0＜p＜1),hereΩis homogeneous of degree zero on Rⁿ satisfying a kind of Dini-type condition.In the second chapter,we consider the boundedness of Marcinkiewicz integral operatorμΩon weighted Herz space K_q^a,p(ω₁;ω₂),and obtain that the Marcinkiewicz integral operatorμΩis bounded on K_q^a,p(ω₁;ω₂), whereω₁,ω₂ is A₁ weight.Letμ_Ω,b^m be the higher order commutator generated by Marcinkiewicz integralμΩand a BMO function b(x).In the third chapter,we will study the continuity ofμΩandμ_Ω,b^m on homogeneous Morrey-Herz space.In the last chapter,we turn to discuss parameter Marcinkiewicz integral operatorμ_Ω^ρ.IfΩ∈Lip_α(S^n-1)is a homogeneous function of degree zero,thenμ_Ω^ρis an operator of type(H^p,∞,L^p,∞)(0＜p≤1). For p=1,we weaken the smoothness condition assumed onΩand again obtainμ_Ω^ρis of type(H^1,∞,L^1,∞).As a corollary of the results above,we give the weak type(1,1)boundedness ofμ_Ω^ρ.