The Study on Boundedness for Multilinear Commutator of Littlewood-paley Operator
|School||Changsha University of Science and Technology|
|Keywords||Littlewood-Paley operator Multilinear commutator BMOspace Herz space Morrey-Herz space Trebel-Lizorkin space A1-weight Weighted Lipschitz space Besov spaces|
In this paper,we study the boundedness of multilinear commutators (?)genierated by Littlewood-Paley operator Sδand locally integrable funections.At first,the (Lp,Lq)-boundedness for the multilinear commutator (?) is proved. In this section,we adopt two methods in proving it. On the one hand,the Sharp funcction inequality is proved,by using it,we obtain (?)is bounded on generalized Morrey spaces. On the other hand,by using the (Lp,Lq)-boundedness for the multilinear commutator (?)the Good-λinequality for the multilinear commutator (?) is proved,where b=(b1,…bm),bj∈BMO(Rn),1≤j≤msecondly,the BMO estimates for the multilinear commutator (?) are proved.In this chapter,we contain two parts.For one thing,we establishλ-central BMO estimates for (?)Oni central Morrey spaces.For another thing,we prove CBMO estimates for the multilinear commutator on Herz and Morrey-Herz spaces,where b=(b1,…bm), bj∈BMO(Rn),1≤j≤m.Thirdly.we get that (?) is bounded from Lp(w)to Lq(w1-q(m-δ/n)),where w A1,0<δ<n,1<p<n／(δ+mβ),1/p-1/q=(δ+mβ)/n,and Lp(w) to (?)(w1-qm(1-(δ-β)/n)),where w∈A1,0<δ<n,1<p<n/mδ,1/p-1/q mδ／n. Where (?) is genierated by the Littlewood -Paley operator and functions in weighted Lipschitz space.Finally,we prove the the boundedness for the multilinear commutator (?) on the Besov spaces.For bj∈Δβ(Rn),then (?) is bounded from Lp(Rn) toΔ(δ+mβ)n/p(Rn) for any n/(mβ+δ)≤p≤n/δ,and (?) is bounded from (?)(Rn)to CL-α/n-1/q2,q2(Rn) with the appropriate condition.