The Boundedness Of Multilinear Commutators Of Marcinkiewicz Operator

In this paper, we study the boundedness of multilinear Marcinkiewicz commu-tators generated by Marcinkiewicz operator and locally integrable functions. Westudy system the boundedness of multilinear commutatorsμδb(0 <δ< n), gener-ated by Marcinkiewicz operatorμδand BMO functions or Lipschitz functions onLp(1 < p <∞) space,Hardy space,Herz -Hardy space,Triebel-Lizorkin space.Moreover, we consider some kind of endpoint estimates for multilinear Marcinkiewiczcommutators.At first, the sharp inequalities for multilinear Marcinkiewicz commutatorsμδbare proved. By using it, we obtainμδb are bounded on Lp space,where 1 < p <∞.After that, the boundedness for multilinear Marcinkiewicz commutatorsμδb onHbp (Rn), H K˙qα,,bp (Rn) and HKqα,,bp (Rn) are proved, where bi∈BMO(Rn),1≤i≤m,b = (b1,···,bm).Then, we get they are bounded from Lp(Rn) to F˙qmβ,∞(Rn),Lp(Rn) to Lq(Rn),where 1/p - 1/q = mβ+δ/n and 1/p > mβ+δ/n, Hp(Rn) to Lq(Rn), H K˙qα1, p(Rn) toK˙qα2, p, H K˙qn1( 1-1/q1)+ε,p(Rn) to W K˙qn2( 1-1/q1)+ε,p(Rn), which generated by Marcinkiewiczoperator and functions in Lipschitz space.Finally, the weighted endpoint estimates for multilinear Marcinkiewicz commu-tatorsμbδare studied. They are bounded from Ln/δto BMO(Rn). Moreover, let0 <δ< n, 1 < p < n/δand b = (b1,···,bm) with bj∈BMO(Rn) for 1≤j≤m. Thenμδb is bounded from Bpδ(Rn) to CMO(Rn).Last, Let 0 <δ< n and b = (b1,···,bm)with bj∈BMO(Rn) for 1≤j≤m. If for any H1(Rn)-atom a supported on certaincube Q and u∈Q, there isthenμbδis bounded from H1(Rn) to Ln/(n-δ)(Rn).