Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

摘要

Let T be a singular integral operator with its kernel satisfying |K(x − y) − ∑k=1 ^ℓ‍B k(x)ϕ k(y)| ≤ C | y|^γ/|x − y|^n+γ, |x | > 2 | y | > 0, where B k and ϕ k  (k = 1,…, ℓ) are appropriate functions and γ and C are positive constants. For b→=(b1,…,bm) with b j ∈ BMO(ℝⁿ), the multilinear commutator Tb→ generated by T and b→ is formally defined by Tb→f(x)=∫ℝn∏j=1m‍(bj(x)-bj(y))K(x,y)f(y)dy. In this paper, the weighted L ^p-boundedness and the weighted weak type Llog⁡L estimate for the multilinear commutator Tb→ are established.