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

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 \(T_{\alpha ,m,i}^b\,(i = 1, \ldots m)\) $$T_{\alpha ,m,i}^b(i=1,\dots m)$$ on product generalized mixed Morrey spaces \(M_{\overrightarrow {{q_1}} }^{{\varphi _1}}({\mathbb{R}^n}) \times \cdots \times M_{\overrightarrow {{q_m}} }^{{\varphi _m}}({\mathbb{R}^n})\) $$M_{\overrightarrow{q_1}}^{{\phi }_1}({ℝ}^n)\times \cdots \times M_{\overrightarrow{q_m}}^{{\phi }_m}({ℝ}^n)$$ . We find sufficient conditions on (ϕ₁, …, ϕ_m, ϕ) which ensure the boundedness of T_α,m from \(M_{\overrightarrow {{q_1}} }^{{\varphi _1}}({\mathbb{R}^n}) \times \cdots \times M_{\overrightarrow {{q_m}} }^{{\varphi _m}}({\mathbb{R}^n})\) $$M_{\overrightarrow{q_1}}^{{\phi }_1}({ℝ}^n)\times \cdots \times M_{\overrightarrow{q_m}}^{{\phi }_m}({ℝ}^n)$$ to \(M_{\vec p}^\varphi ({\mathbb{R}^n})\) $$M_{\overrightarrow{p}}^{\phi }({ℝ}^n)$$ . Moreover, we also give sufficient conditions for the boundedness of T_α,m,i from \(M_{\overrightarrow {{q_1}} }^{{\varphi _1}}({\mathbb{R}^n}) \times \cdots \times M_{\overrightarrow {{q_m}} }^{{\varphi _m}}({\mathbb{R}^n})\) $$M_{\overrightarrow{q_1}}^{{\phi }_1}({ℝ}^n)\times \cdots \times M_{\overrightarrow{q_m}}^{{\phi }_m}({ℝ}^n)$$ to \(M_{\vec p}^\varphi ({\mathbb{R}^n})\) $$M_{\overrightarrow{p}}^{\phi }({ℝ}^n)$$ . 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.