Boundedness For A Class Of Marcinkiewicz Integral Operators

In this dissertation, the boundedness for a class of Marcinkiewicz Integrals are mainly considered.In the first chapter,we give the (Lp(v), LP(u)) boundedness for Marcinkiewicz integral operator uΩ related to the Littlewood-paley g-function . Moreover,the (LP(v), Lp(u)) boundedness for Marcinkiewicz integral operators u*Ωλ, and uΩ,S related to the Littlewood-paley g-λfunction and the Lusin area integral S are obtained. As application ,we prove that the (Lp(v),Lp(w)) boundedness for the higher-order commutators u*Ω,b, u*Ωmλ,b, umΩ,S,b which are formed by these Marcinkiewicz integral operators and a BMO function.In the second chapter, we consider the parametric Marcinkiewicz integrals upΩ and obtain it is the operator of type (Hp, Lp), (0 < p < 1). Here Ω satisfies some smoothness conditions.In the third chapter,the boundedness properties are considered for a class of sublinear operators,from weighted Herz space Knq(1-1/q),p(w1;w2) to weighted weak Herz space WKnq(1-1/q),p(w1;w2) .In particular,the boundedness for Marcinkiewicz integral uΩ from weighted Herz space Knq(1-1/q),p(w1;w2) to weighted weak Herz space WKnq(1-1/q),p(w1;w2) are established.