Some Estimates For Marcinkiewicz Type Operators And Its Commutators

There are three chapters in this thesis,which focuses on three contents:the bounded-ness of the Marcinkiewicz integral operator on certain weighted Campanato spaces; thebounededness of the fractional type Marcinkiewicz integral; estimates for the fractionaltype Marcinkiewicz Integral Commutator.The outline of the paper is arranged as follows:In Chapter 1,we obtain the boundedness of the Marcinkiewicz integral operator oncertain weighted Campanato spaces,which generalize the results that we know.Suppose that S^n-1 is the unit sphere of Rn(n≥2)equipped with the normalizedLebesgue measure dσ= dσ(x). LetΩbe homogeneous of degree zero and satisfies thecancelation conditionwhere x = x/|x| for any x = 0. We define the fractional type Marcinkiewicz integral ofhigher dimension bywhereIn chapter 2,we obtain the boundedness of the fractional type Marcinkiewicz Integralμ_Ω,α with the kernel satisfying L^q-Dini condition.Let b(x)∈Lloc(Rn), then the fractional type Marcinkiewicz integral commutator ofhigher dimension is defined bywhere In chapter 3,we obtain estimates for the fractional type Marcinkiewicz Integral Com-mutatorμ_b^Ω,α on Hardy spaces with the kernel satisfying the logarithmic type Lipschitzcondition, where b∈Lip_α(0 <α≤1).