Boundedness Criterion for Multi-sublinear Operators and Their Commutators Generated by Multilinear Fractional Integral Operator on Product Generalized Mixed Morrey Spaces

Abstract

The purpose of this paper is to study the boundedness for a large class of multi-sublinear operators T_α,m generated by multilinear fractional integral operator and their commutators $T_{\alpha ,m,i}^b\,(i = 1, \ldots m)$ on product generalized mixed Morrey spaces $M_{\overrightarrow {{q_1}} }^{{\varphi _1}}({\mathbb{R}^n}) \times \cdots \times M_{\overrightarrow {{q_m}} }^{{\varphi _m}}({\mathbb{R}^n})$. We find sufficient conditions on (ϕ₁, …, ϕ_m, ϕ) which ensure the boundedness of T_α,m from $M_{\overrightarrow {{q_1}} }^{{\varphi _1}}({\mathbb{R}^n}) \times \cdots \times M_{\overrightarrow {{q_m}} }^{{\varphi _m}}({\mathbb{R}^n})$ to $M_{\vec p}^\varphi ({\mathbb{R}^n})$. Moreover, we also give sufficient conditions for the boundedness of T_α,m,i from $M_{\overrightarrow {{q_1}} }^{{\varphi _1}}({\mathbb{R}^n}) \times \cdots \times M_{\overrightarrow {{q_m}} }^{{\varphi _m}}({\mathbb{R}^n})$ to $M_{\vec p}^\varphi ({\mathbb{R}^n})$. As applications, the boundedness for multi-sublinear fractional maximal operator, multilinear fractional integral operator and their commutators on product generalized mixed Morrey spaces is established.