Boundedness Of The Commutator Of Marcinkiewicz Integral Operators Boundedness Problems

In the present thesis, we mainly study the boundedness of Marcinkiewicz integral operators on the weighted Herz spaces,which consists of two chapters.In the first, chapter, we discuss the-boundedness of Marcinkiewicz integral operators with rough kernels on the weighted Herz spaces, and get the following theorem:Theorem 1.1.1 Let Ω∈ L^T(Sⁿ¹) for some r ∈ (1,∞],0 < p < ∞, 1 < q < ∞,w₁∈A₁,w₂ ∈ A₁ and w₂ satisfies(1.1.3). If q > r' and a G (—n/q, n(1/r' —1/q) + 1/r),then μ_Ω is bounded on K_p^a,p(w₁;w₂).In the second chapter, we mainly discuss the boundedness of Marcinkiewicz integral operators with rough kernel on the weighted Herz-type Hardy spaces, and have the following theorem:Theorem 2.1.1 Let 01 G A₁,w₂ G A₁ and w₂ satisfies (1.1.3), if there exists r > max{q,q'}(1/q + 1/q' = 1) such that Ω∈ L^r (S^n-1) and satisfiesthen μ_Ω is bounded from HK+q^a,p(w₁;w₂) into K_p^a,p(w₁;W₂).Theorem 2.1.2 Let 1 < q < ∞,0 < p ≤ 1, w₁, w₂ ∈ A₁ and w₂ satisfies (1.1.3), if there exists r > max{q, q'} such that Ω ∈ L^r(S^n-1) and exist 77 satisfy ηp > 1 such thatthen μ_Ω is bounded from intoCorollary 2.1.3 Let 1 < q < oo,Wi,w2 ï¿¡ A\ and w2 satisfies (1.1.3). If there exists r > max{q, q'} such that ft G I/"^?"1) and satisfies Lr—Dini condition (2.1.2),then jj.n is bounded from Hk^{l-l/q)^(wx;W2) into...