A weighted weak-type multilinear gradient inequality

We characterize the weights \(w, v_1, v_2, \dots , v_m \) for which the weak-type multilinear gradient inequality

$$\begin{aligned} \left\| \prod _{i=1}^m f_i\right\| _{p,\infty ;w}\le C \prod _{i=1}^m \left\| x \cdot \nabla f_i(x)\right\| _{p_i,v_i} \end{aligned}$$

holds for all \(f_1, f_2, \dots , f_m \in C_c^{\infty }({\mathbb {R}}^n)\) in the case \(\frac{1}{p} = \frac{1}{p_1}+\frac{1}{p_2}+ \cdots + \frac{1}{p_m}\).