Two-weight Inequalities For Multilinear Singular Integral Operators And Commutators

In this paper,we discuss two-weight inequalities for the multilinear singular integral operators,the multilinear fractional integral operators and commutators.For the multilinear Calderon-Zygrnund singular integral operators Υ,we obtain the following results:let 1<p1,…,pm<∞,1/p=1/p1+…+1/(pm).(1)Let(v,u=(v,v1,...,vm)be a pair of weights for which there exists r>1 such that for all cubes Q, then T is a bounded operator from Lp1(u1)×…×Lpm(um)to Lp,∞(v),(2)Let(v,u)=(v,u1,…,um)be a pair of weights for which there exists r>1 such that for all cubes Q, where Bi(i=1,…,m)is a doubling Young function such that Bi=Bpi(i=1,…,m), then (?) is a bounded operator from Lp1(u1)×...× Lpm(um)to Lp(v).For the commutators of multilinear Calderon-Zygmund operators (?)=(b1,…,bm)∈ BMOm,we obtain the following results:let 1<p1,…,pm<∞,1/p=1/p1+…+1/(pm).(1)Let(v,u)=(v,u1,…,um)be a pair of weights for which exists r>1 such that for all cubes Q, where Ai(t)=tp:log(e+t)p:,then (?) is a bounded operator from Lp1(u1)×...×Lpm(um) to Lp,∞(v).(2)Let Ai and Di(i=1,…,m)are Young functions such that Ai ∈Bpi,and Ai-1(t)Di-1(t)≤B-1(t),where B(t)=t log(e+t),(v,u)=(v,u1,…,um)be a pair of weights for which exists r>1 such that for all cubes Q, then (?) is a bounded operator from Lp1(u1)×…×Lpmi(um)to Lp(v).For the multilinear fractional singular integral operators (?):0<α<mn.we obtain the following results:let 1<p1,…,pm<∞,1/p=1/p1+…+1/(pm),1/q=1/p-a/n,(v,u)= (v,u1,…,um)be a pair of weights for which exists r>1 such that for all cubes Q, then (?)is a bounded operator from Lp1(u1)×…×Lpm(um)to Lp,∞(v).For the commutators of multilinear fractional type operators operators (?),0<α< mn,b=(b1,…,bm)∈BMOm,we obtain the following results:let 1<p1,…,pm<∞, 1/p=1/p1+…+1/(pm),1/q=1/p-a/n,(v,u)=(v,u1,…,um)be a pair of weights for which exists r>1 such that for all cubes Q, where Ai(t)=tpi:log(e+t)pi:then (?) is a bounded operator from Lp1(u1)×…×Lpm(um) to Lp,∞(v).