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

Abstract

We obtain sufficient and necessary conditions on weight functions $s_1(t),\ldots,s_m(t)$ and $\psi(t)$ so that the weighted multilinear Hardy-Ces\`{a}ro operator \[(f_1,\ldots,f_m)\mapsto \int_{[0,1]^n}\left(\prod_{k=1}^nf_k\left(s_k(t) x\right)\right)\psi(t)dt \] 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 $\dot{K}^{\alpha, p}_{q}(\omega)$ and 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 $M\dot{K}^{\alpha, \lambda}_{p,q}(\omega)$. The sharp bounds are also obtained and these results hold for both cases $0<p<1$ and $1\leq p<\infty$. We give a sufficient condition so that if symbols $b_1,\ldots,b_m$ are Lipschitz, then the commutator of the weighted Hardy-Ces\`{a}ro operator \[ (f_1,\ldots,f_m)\mapsto\int_{[0,1]^n}\left(\prod\limits_{k=1}^mf_k\left(s_k(t)x\right)\right)\left(\prod_{k=1}^m\left(b_k(x)-b_k\left(s_k(t)x\right)\right)\right)\psi(t)dt\] 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 $M\dot{K}^{\alpha^\prime, \lambda}_{p, q}(\omega)$ for both cases $0<p<1$ and $1\leq p<\infty$. By these we extend and strengthen previous results deu to Tang, Xue, and Zhou [16].