The Study On Vector-valued Maximal Multilinear Commutators Of Bochner Riesz Operator

In the classical and modern mathematics, function spaces play a very impor-tant role. In the field of harmonic analysis, one soon encounters the Lebesgue space Lp, the Hardy space Hp, Lipschitz space and space BMO. The original definitions of these spaces do not see the close link between them. In 1970's, with a deeper understanding of interpolation theory, homogenous Triebel—Lizorkin spaces Fpβ,∞and Besove spaces Bpβ,∞(or inhomogenous Triebel—Lizorkin spaces Fpβ,∞and Besove spaces Bpβ,∞) which can unify all these function spaces mentioned above came up.On the other hand, the bounded research about operators among above-mentioned spaces has been one of the central issues of harmonic analysis. As an important multiplier operator in harmonic analysis, and because it is closely linked with the sums of trigonometric series, Bochner—Riesz operator interest the analyst all the time and researches on it have achieved a lot. In 2002, Perez and Trujillo-Gonzalee raised the concept of the multilinear commutator of the sin-gular integral operators. The study found many of the results of the commutator of operators have a similar nature with ordinary operators. By this inspiring, in this article, we first propose the concept of vector-valued maximal Bochner—Riesz multilinear commutators |Bδ,*b(f)|r generated by a few special class of the locally integrable functions and the Bochner—Riesz operator and study its boundedness on some functions space.There are three chapters in this thesis.In Chapter 1 we first describes the research background and significance of this article, and introduce the definitions of spaces of Lipschitz spaces and ho-mogenous Triebel—Lizorkin spaces. Also, the definition of vector-valued maxi-mal Bochner—Riesz multilinear commutators |Bδ,*b(f)|r is given. For 1< r<∞, the vector-valued multilinear commutator associated with the maximal Bochner Riesz operator is defined by where Bδ,*b(f)(x)=(?) |Bδ,tb(f)(x)|, and Btδ(z) = t-n Bδ(z/t) for t > 0.Let H be the space H = {h:‖h‖= (?) |h(t)| <∞}, then it is clear that Bδ,*b(f)(x)=‖Bδ,tb(f)(x)‖and Bδ*(f)(x) =‖Btδ(f)(x)‖. SetIn chapterⅡ, we mainly study the boundedness of the vector-valued multilinear commutator of maximal Bochner - Riesz operator |Bδ,*b(f)|r on Lp(w), where 1 < p <∞, w∈Ap. To obtain this result, we conducted a sharp function estimate for |Bδ,*b(f)|r and obtainTheorem 0.0.1 Let 1 < r <∞,δ> (n- 1)/2, bj∈BMO(Rn), where 1≤j≤m, m∈N. Then for any 1 < s <∞, there is a constant C > 0, for all f∈C0∞(Rn),x∈Rn, After that, we get the result by the induction. At the same time, we gain elicitation from the process of the prove, we found that |Bδ,*b(f)|r is bounded from L∞(w) to BMO(w), which is an endpoint estimate for the vector-valued multilinear commu-tator, and we haveTheorem 0.0.2 Let 1 < r <∞,δ> (n - 1)/2, bj∈BMO(Rn), where 1≤j≤m, m∈N. Then |Bδ,*b(f)|r is bounded on Lp(w), where w∈Ap, 1 < p <∞.Theorem 0.0.3 If 1 < r <∞,δ> (n- 1)/2, w∈A1, bj∈BMO(Rn), 1≤j≤m, m∈N, then |Bδ,*b(f)|r is bounded from L∞(w) to BMO(w).In the following, we study the vector-valued multilinear commutator |Bδ,*b(f)|r which is generated by some functions in Lipβ(Rn) and maximal Bochner - Riesz operator. We get the boundedness of the vector-valued multilinear commutator |Bδ,*b(f)|r on Lebesgue and Triebel - Lizorkin space in last part. Our main theo-rems are Theorem 0.0.4 Let 1 < r <∞,δ> (n-1)/2, m∈N, 0 <β< min(1/m, (2δ- n + 1)/2m), 1 < p <∞, b= (b1,...,bm), where bj∈Lipβ(Rn), 1≤j≤m. Then |Bδ,*b|r is bounded from Lp(Rn) to Fpmβ,∞(Rn)Theorem 0.0.5 Let 1 < r <∞,δ> (n-1)/2, m∈N, 0 <β< min (1/m,(2δ-n+1)/2m),1 < p∞,b = (b1,...,bm), where bj∈Lipβ(Rn), 1≤j≤m. Then |Bδ,*b|r is bounded from Lp(Rn) to Lq(Rn), where 1/p - 1/q mβ/n, 1/p > mβ/n...