Endpoint estimates for the commutators of multilinear Calder\'{o}n-Zygmund operators with Dini type kernels

Let $T_{\vec{b}}$ and $T_{\mathrm{\Pi }b}$ be the commutators in the $j$-th entry and iterated commutators of the multilinear Calder'{o}n-Zygmund operators, respectively. It was well-known that $T_{\vec{b}}$ and $T_{\mathrm{\Pi }b}$ were not of weak type $(1,1)$ and $(H^1,L^1)$, but they did satisfy certain endpoint $L\log L$ type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that $T_{\vec{b}}$ and $T_{\mathrm{\Pi }b}$ are bounded from product Hardy space $H^1\times \cdot \cdot \cdot \times H^1$ to weak $L^{\frac{1}{m},\mathrm{\infty }}$ 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.