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{ó}n-Zygmund operators and their commutators $T_{m,i}^b(i=1,\cdots ,m)$ on the product generalized mixed Morrey spaces $M_{\overrightarrow{q_1}}^{{\phi }_1}({\mathbb{R}}^n)\times \cdots \times M_{\overrightarrow{q_m}}^{{\phi }_m}({\mathbb{R}}^n)$. We find the sufficient conditions on $({\phi }_1,\cdots ,{\phi }_m,\phi )$ which ensure the boundedness of the operator $T_m$ from $M_{\overrightarrow{q_1}}^{{\phi }_1}({\mathbb{R}}^n)\times \cdots \times M_{\overrightarrow{q_m}}^{{\phi }_m}({\mathbb{R}}^n)$ to $M_{\vec{q}}^{\phi }({\mathbb{R}}^n)$. Moreover, the sufficient conditions for the boundeness of $T_{m,i}^b$ from $M_{\overrightarrow{q_1}}^{{\phi }_1}({\mathbb{R}}^n)\times \cdots \times M_{\overrightarrow{q_m}}^{{\phi }_m}({\mathbb{R}}^n)$ to $M_{\vec{q}}^{\phi }({\mathbb{R}}^n)$ are also studied. As applications, we obtain the boundedness for the multi-sublinear maximal operator, the multilinear Calder{ó}n-Zygmund operator and their commutators on product generalzied mixed Morrey spaces.