Multi-sublinear operators and their commutators on product generalized mixed Morrey spaces

Abstract: In this paper, we study the boundedness for a large class of multi-sublinear operators $T_m$ generated by multilinear Calder{\'o}n-Zygmund operators and their commutators $T^{b}_{m,i}~(i=1,\cdots,m)$ on the product generalized mixed Morrey spaces $M^{\varphi_1}_{\vec{q_1}}({\Bbb R}^n)\times\cdots\times M^{\varphi_m}_{\vec{q_m}}({\Bbb R}^n)$. We find the sufficient conditions on $(\varphi_1,\cdots,\varphi_m,\varphi)$ which ensure the boundedness of the operator $T_m$ from $M^{\varphi_1}_{\vec{q_1}}({\Bbb R}^n)\times\cdots\times M^{\varphi_m}_{\vec{q_m}}({\Bbb R}^n)$ to $M^{\varphi}_{\vec{q}}({\Bbb R}^n)$. Moreover, the sufficient conditions for the boundeness of $T^b_{m,i}$ from $M^{\varphi_1}_{\vec{q_1}}({\Bbb R}^n)\times\cdots\times M^{\varphi_m}_{\vec{q_m}}({\Bbb R}^n)$ to $M^{\varphi}_{\vec{q}}({\Bbb R}^n)$ are also studied. As applications, we obtain the boundedness for the multi-sublinear maximal operator, the multilinear Calder{\'o}n-Zygmund operator and their commutators on product generalzied mixed Morrey spaces.