Commutators Of Marcinkiewicz Integral Boundedness

Let Cb(f) be the commutator which is formed by the Marcinkiewicz integral μ(f) and a function b ∈ Aβ . We discuss the boundedness of the Marcinkiewicz commutator from Lp to L1(1 < p < ?), the weak type L1 boundedness and the boundedness from Hp to Lq(n/(n+ β) < p^1) in chapter two, chapter three and chapter four, respectively,then we get the following conclusions:Theoreml Let Ω(x) be homogeneous of degree zero, satisfying the condition (a),(b), and 1 < p < ?, 0 < β< 1,1/p =1/q -∈/n. If b ∈ β, thenTheorem2 Let Ω(x) be homogeneous of degree zero, satisfying the condition Ls-Dini. If b ∈ β(0 < β < 1), then Cb is bounded from Hn/n+β(Rn) to L1(Rn).Theorems Let Ω(x) be homogeneous of degree zero, satisfying the condition (a),(b). If b ∈ β(0 < β < 1), then for n/(n + β] < p ≤1, there exist a constant C(C > 0), independent of f, such that||Cb(f)||Lq ≤C||f||HP ||b|| β...