Abstract
This note concerns multiple weighted inequalities for vector-valued multilinear singular integral operator with nonsmooth kernel and its corresponding commutators containing multilinear commutator and iterated commutator generated by the vector-valued multilinear operator and BMO functions. By the weighted estimates for a class of new variant maximal and sharp maximal functions, the multiple weighted norm inequalities for such operators are obtained.
Highlights
It is well known that multiple weighted norm inequalities for multilinear operators and their related commutators on various spaces of function is a center topic of harmonic analysis, which recently attracts a lot of attention, see [1,2,3] et al In this paper, we will focus on the multiple weighted estimates for vector valued multilinear singular integral with nonsmooth kernel and its commutators
We give some information on multilinear Calderon-Zygmund operators
The multilinear operator T we study is initially defined on the m-fold product of Schwartz space S(Rn)×⋅ ⋅ ⋅×S(Rn) and taking values into the space of tempered distributions S(Rn); that is, T : S (Rn) × ⋅ ⋅ ⋅ × S (Rn) → S (Rn)
Summary
It is well known that multiple weighted norm inequalities for multilinear operators and their related commutators on various spaces of function is a center topic of harmonic analysis, which recently attracts a lot of attention, see [1,2,3] et al. It is natural to ask whether the vectorvalued multilinear operator Tq with kernels satisfying the same conditions as in [5] and its commutators TΣb⃗,q and TΠb⃗,q have multiple weighted estimates or not. These problems will be addressed by our theorems. Assume that Tq is a vector-valued multilinear operator defined as (10) associated with T being an m-GCZK(A, s, η, ε) whose kernel K satisfies Assumption (H2). Assume that Tq is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the iterated commutator TΠb⃗,q is defined as (13). Throughout this paper, we use the letter C to denote a positive constant that varies from line to line, but it is independent of the essential variable
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