| In the article.we mainly study the boundedness for the multilinear commuta-tor related to the generalized singular integral operator and some locally integrable functions on Rn. In this paper, we research the boundedness of the multilinear com-mutators Tb generated by the generalized singular operator T and BMO functions or weighted Lipschitz functions in Lp(1<p<∞) spaces、Besov space, we also study the certain generalized fractional integral operator.At first, the weighted Lp-boundedness for the multilinear commutator Tb is proved. In this section, we prove a sharp function inequality firstly. By using it, we obtain Tb is bounded from Lp(w) to Lq(w) as well as from Tp,φ(w) to Lp,φ(w), where1<p<∞and w∈Ap. Later, we prove the Mk-type inequality for the multilinear commutator.Secondly, we get the weighted estimates for the multilinear commutator Tb re-lated to the generalized singular integral operator and the weighted Lipschitz func-tions. We obtain the boundedness of the multilinear commutator Tb from Lp(w) to Lr(w1-m+(r-1)(mβ)/n)and from Lp(w) to Fpm β,∞(w1-m-(mβ)/n). In the discussion of bound-edness from Lp(w) to Lr(w1-m+(r-1)(mβ)/n. the corresponding indicators should satisfies bj∈Lipβ(w),1≤j≤m,0<β<1and w∈A1. and in the situation from Lp(w) to Fpmβ,∞(w1-m-(mβ)/n), they should satisfy bj∈Lipβ(w),1≤j≤m,.0<β<1and w∈A1.Thirdly, we prove the the boundedness for the multilinear commutator Tb on Rn. In this chapter, we also give the related proof according two aspects. For one thing, we obtain that Tb is bounded from Lp(Rn) to△(mβ)/n-1/p(Rn), where0<β<1/m,(q’<p<n/mβ and bj∈△β(Rn) for j=1,…,m;And for another, we prove Tb is boundedFinally, we prove some sharp maximal function estimates for the commutators related to certain generalized fractional integral operator with general kernel and the BMO and Lipschitz functions. As an application, we obtain the boundedness of the commutators on Lebesgue,Alorrey and Triebel-Lizorkin spaces. |
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