Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels

Abstract

Abstract Let T b → $T_{\vec{b}}$ and T Π b $T_{\Pi b}$ be the commutators in the jth entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well known that the commutators of linear Calderón-Zygmund operators were not of weak type ( 1 , 1 ) $(1,1)$ and ( H 1 , L 1 ) $(H^{1}, L^{1})$ , but they did satisfy certain endpoint L log L $L\log L$ type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that T b → $T_{\vec{b}}$ and T Π b $T_{\Pi b}$ are bounded from the product Hardy space H 1 × ⋯ × H 1 $H^{1}\times\cdots\times H^{1}$ to weak L 1 m , ∞ $L^{\frac{1}{m},\infty}$ space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by Christ, a very complex decomposition of the integrand domains, and carefully splitting the commutators into several terms.

Keywords

• commutators
• multilinear Calderón-Zygmund operator
• C-Z kernel of ω type
• Dini type conditions
• Hardy spaces