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

2016/01/10 by Li, Zhengyang, Xue, Qingying
#42B25 #47G10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1601.02173

Abstract

Let T_b and T_Πb be the commutators in the j-th entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well-known that T_b and T_Πb were not of weak type (1,1) and (H¹, L¹), but they did satisfy certain endpoint Llog L type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that T_b and T_Πb are bounded from product Hardy space H¹×⋅⋅⋅× H¹ to weak L^(1)/(m),∞ space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by M. Christ, a very complex decomposition of the integrand domains and splitting and estimating the commutators very carefully into several terms and cases.

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