Multilinear Hardy-Ces\`aro Operator and Commutator on the product of Morrey-Herz spaces

We obtain sufficient and necessary conditions on weight functions $s_1(t),\dots ,s_m(t)$ and $\psi (t)$ so that the weighted multilinear Hardy-Ces`{a}ro operator $$(f_1,\dots ,f_m)\mapsto {\int }_{[0,1{]}^n}\left(\prod _{k=1}^n{f}_k(s_k(t)x)\right)\psi (t)dt$$ is bounded from $\dot{K}^{\alpha_1, p_1}{q_1}(\omega_1)\times \cdots \times\dot{K}^{\alpha_m, p_m}{q_m}(\omega_m)$to$\dot{K}^{\alpha, p}{q}(\omega)$andfrom$M\dot{K}^{\alpha_1, \lambda_1}{p_1,q_1}(\omega_1)\times \cdots \times M\dot{K}^{\alpha_m, \lambda_m}{p_m,q_m}(\omega_m)$to$M\dot{K}^{\alpha, \lambda}{p,q}(\omega)$. The sharp bounds are also obtained and these results hold for both cases $0<p<1$ and $1\le p<\mathrm{\infty }$. We give a sufficient condition so that if symbols $b_1,\dots ,b_m$ are Lipschitz, then the commutator of the weighted Hardy-Ces`{a}ro operator $$(f_1,\dots ,f_m)\mapsto {\int }_{[0,1{]}^n}\left(\prod _{k=1}^m{f}_k(s_k(t)x)\right)\left(\prod _{k=1}^m(b_k(x)-b_k(s_k(t)x))\right)\psi (t)dt$$ is bounded from $M\dot{K}^{\alpha_1, \lambda_1}{p_1, q_1}(\omega_1)\times \cdots \times M\dot{K}^{\alpha_m, \lambda_m}{p_m, q_m}(\omega_m)$to$M\dot{K}^{\alpha^\prime, \lambda}_{p, q}(\omega)$forbothcases$0<p<1$and$1\leq p<\infty$. By these we extend and strengthen previous results deu to Tang, Xue, and Zhou [16].