The author treats the multilinear singular integral operators of the form \(T(f_1,\ldots,f_m)(x)=\int_{\mathbb R^n}\cdots\int_{\mathbb R^n}K(x,y_1,\dots,y_m) f_1(y_1)\cdots f_m(y_m)\,dy_1\cdots dy_m\), where \(K\) satisfies \(| K(x,y_1,\dots,y_m)| \leq A\bigl(\sum_{i=1}^{m}| x-y_i| \bigr)^{-nm}\). This class of operators include the \(m\)-linear Calderón-Zygmund singular integral operators treated by Grafakos and Torres. One of the author's results is as follows: Let \(01\) and \(T\) continuously maps \(L^{q_1}(\omega_{21}^{q_1})\times \cdots\times L^{q_m}(\omega_{2m}^{q_m})\) into \(L^q(\omega_2^q)\), it follows that \(T\) also continuously maps the homogeneous weighted Herz space \(\dot K_{q_1}^{\alpha_1, p_1}(\omega_1, \omega_{21}^{q_1})\times \cdots\times \dot K_{q_m}^{\alpha_m, p_m}(\omega_1, \omega_{2m}^{q_m})\) into \(\dot K_{q}^{\alpha, p}(\omega_1, \omega_2^q)\), provided \(-nD_i/q_i<\alpha_iq(\omega_i)