The Boundedness Of Vector-Valued Multilinear Commutators For Marcinkiewicz Operator

Marcinkiewicz integral operator commutator is important in harmonic anal-ysis operator. It is well-known that the Littlewood - Paley-g function was intro-duced when Littlewood and Paley researched series. The Littlewood-Paley-g function is within the unit circle through the given, Although Marcinkiewicz op-erator have a similar nature with Littlewood-Paley-g function, It is no longer needed within the unit circle is given by. This is the hope that people looking to replace Littlewood-Paley-g function. Marcinkiewicz integral operator, not only in harmonic analysis plays an important role, and in the application of partial differential equation is particularly true. Many scholars have studied the problem of the boundedness for Marcinkiewicz operator in some function spaces for a long time, and they are still studying this problem further.In this paper, we study the boundedness in some function spaces of the vector-valued multilinear commutators generated by Marcinkiewicz operator and locally integrable functions. Moreover, we also consider some kind of endpoint estimates for the vector-valued multilinear Marcinkiewicz commutators.There are four chapters in this paper.The first chapter mainly introduces the background and singnificance about the vector-valued multilinear Marcinkiewicz commutators (?), the common symbols and preliminaries.In the second chapter, the sharp function inequalities for the vector-valued multilinear Marcinkiewicz commutators (?) are proved. By using it, we obtain that (?) are bounded on Lp space, where 1< p<∞.In the third chapter, we prove that (?) are bounded from Lp(Rn) to Fpmβ,∞(Rn), Lp(Rn) to Lq(Rn), where 1/p-1/q=mβ/n and 1/p>mβ/n, which is generated by Marcinkiewicz operator and the functions in Lipschitz space.In the last chapter, the weighted endpoint estimates for the vector-valued multilinear Marcinkiewicz commutators (?) are studied. They are bounded from L∞(w) to BMO(w). Moreover, if w(Q)≥1 for any cube Q and t> max{p, s},λ≤0, then (?) are bounded from Bt,λ(w) to CMOs,λ(w), where w∈Ap,(p>1).