The Study on Boundedness for Multilinear Commutator of Littlewoodpaley Operator 

Author  ShenYing 
Tutor  HuangChuangXia 
School  Changsha University of Science and Technology 
Course  Applied Mathematics 
Keywords  LittlewoodPaley operator Multilinear commutator BMOspace Herz space MorreyHerz space TrebelLizorkin space A1weight Weighted Lipschitz space Besov spaces 
CLC  O177 
Type  Master's thesis 
Year  2012 
Downloads  10 
Quotes  0 
In this paper,we study the boundedness of multilinear commutators (?)genierated by LittlewoodPaley 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 Goodλinequality 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 establishλcentral BMO estimates for (?)Oni central Morrey spaces.For another thing,we prove CBMO estimates for the multilinear commutator on Herz and MorreyHerz spaces,where b=(b1,…bm), bj∈BMO(Rn),1≤j≤m.Thirdly.we get that (?) is bounded from Lp(w)to Lq(w1q(mδ/n)),where w A1,0<δ<n,1<p<n／(δ+mβ),1/p1/q=(δ+mβ)/n,and Lp(w) to (?)(w1qm(1(δβ)/n)),where w∈A1,0<δ<n,1<p<n/mδ,1/p1/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 CLα/n1/q2,q2(Rn) with the appropriate condition.