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

Abstract

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 trueb→=false(b1,…,bmfalse) with b j ∈ BMO(ℝn), the multilinear commutator Tb→ generated by T and b→ is formally defined by Ttrueb→ffalse(xfalse)=∫ℝn[]|false∏j=1mnormal‍false(bjfalse(xfalse)-bjfalse(yfalse)false)Kfalse(x,yfalse)ffalse(yfalse)dy. In this paper, the weighted L p-boundedness and the weighted weak type Llog⁡L estimate for the multilinear commutator Tb→ are established.