The Boundedness Of Bochner-Riesz Maximal Multilinear Commutators

In this paper, we study the boundedness of maximal operator associated withthe multilinear commutator of Bochner Riesz operator, where bj belongs toBMO(Rn) or Lipβ(Rn). The maximal operator Bδb, associated with the multilinearcommutator of the Bochner Riesz operator is defined bywhereand Btδ(z) = tnBδ(z/t), t > 0. We still get some endpoint estimates as thesupplement of main theorems.First, we study the continuity of the maximal multilinear commutator ofBochner Riesz operator Bδb, on Lp(w), where w is a weight. We get the sharpestimate of Bδb, in order to obtain the continuity, this is the key of the proof.After that, we use the induction, we will get the result. At the same time, we gainelicitation from the process of the prove, we found that Bb,δis bounded from L∞to BMO(w), which is a endpoint estimate of Bb,δ.Following that, we will investigate the continuities of Bδb, on Hbp (Rn), K˙qα,p(Rn)and H K˙qα,,bp (Rn). Hbp (Rn) and H K˙qα,,bp (Rn) just are the subspace of Hp(Rn) andH K˙qα,p(Rn) respectively, whereas, Bδb, isn't continuous on two classical spaces. Inorder to make the estimate more accurate, we obtain two endpoint estimates ofBδb, on weak Hardy space and prove the continuity of Bδb, on H K˙qα,p(Rn) whichis in the endpoint case.The maximal multilinear commutator Bδb, which we just studied is generatedby m BMO functions and Bochner Riesz operator. We will study maximalmultilinear commutator Bδb, which is generated by some functions in Lipβ(Rn)and Bochner Riesz operator, the definitions of these two kinds of commutatorsare same. We get the continuities of the maximal multilinear commutator Bδb, onLebesgue space, Triebel Lizorkin space, Hardy space, Herz Hardy space inlast part. To obtain those results, we prove them in the same way as before. Weshould find second kind maximal multilinear commutator Bδb, have more propertiesfrom the proofs, we can have the conjecture that most properties of first kindcommutator are properties of second kind commutator. These results indicate the properties of maximal multilinear commutator Bδb,profoundly and reveal a fact that the di?erence between the properties of di?er-ent kinds of commutators has relation with the continuity of locally integrablefunctions bjs.