New endpoint estimates for commutators of multilinear operators

Let $b\in \mathrm{BMO}$ and $T_{\alpha }$ be a class of multilinear operators as Definition 1.3. We establish a new weighted endpoint estimate of the commutators $[\overrightarrow{b},T_{\alpha }]$ via extending the weighted LlogL type Orlicz space to the larger Ω type Orlicz space. As applications, we first obtain the new weighted endpoint estimates for multilinear commutators and iterated commutators associated with multilinear Calderón–Zygmund operators and multilinear fractional integral operators, which improve [26, Theorem 1.1] in some sense. In addition, the new weighted endpoint estimates of commutators associated with multilinear Littlewood-Paley type operators and multilinear singular integrals with non-smooth kernels are shown by enhancing the conditions of operator and their commutator. In particular, the new weighted endpoint estimates are also new results in the one-linear and unweighted cases. Furthermore, we get the endpoint estimate of commutators $[b,T_0]$ on generalized Orlicz spaces by an extrapolation theorem.