| It is well known that Harmonic analysis has been one of the core research areas ofmodern mathematics, and is widely used in partial diferential equations. The central re-search contents of Harmonic analysis are the theory of function spaces and operators withtheir applications. Marcinkiewicz integrals are one of the most important operators in Har-monic analysis. Many mathematicians have researched the properties of Marcinkiewiczintegral operators and the parametric Marcinkiewicz integral operators in diferent func-tion spaces, for example, Lp, BMO, Hp. etc. On the basis of the previous achievements,this paper promote to discuss certain endpoint estimates and weighted endpoint estimatesfor the parametric Marcinkiewicz integral commutators. Our results will enrich the theoryof parametric Marcinkiewicz integral commutators. This paper is organized as follows:In section1, we will introduce the background and current research situation forMarcinkiewicz integral operators and the parametric Marcinkiewicz integral operatorsand state my main work.In section2, we show that the higher order commutators of parametric Marcinkiew-cicz integrals are bounded from L∞（ω） to BMO（ω） with Ω∈L∞(Sn-1), moreover if thekernel also satisfes a logarithm style of Lipshitz condition （1.1）, it maps from Bp（ω） toC MO（ω）.In the next section, the boundedness of the multilinear commutators of parametricMarcinkiewcicz integrals from L∞（ω） to BMO（ω） with Ω∈L∞(Sn-1) is obtained, mean-while the boundedness from Bp（ω） to C MO（ω） is also valid when Ω∈Lipγ(Sn-1)∩L∞(Sn-1).In the last section, if Ω satisfes a Lq-Dini condtion, we can prove that the higherorder commutators of parametric Marcinkiewcicz integrals are bounded from Hbm1（Rn） toL1（Rn）. |
