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,\dots ,m) on product generalized mixed Morrey spaces M_{{\overrightarrow{q}}_1}^{{\varphi }_1}\left({ℝ}^n\right)\times ...\times M_{q_m\to }^{{\varphi }_m}\left({ℝ}^n\right). We find sufficient conditions on (ϕ₁, ..., ϕ_m, ϕ) which ensure the boundedness of T_α,m from M_{\overrightarrow{q_1}}^{{\varphi }_1}\left({ℝ}^n\right)\times ...\times M_{q_m\to }^{{\varphi }_m}\left({ℝ}^n\right)\text{to}{M}_{\overrightarrow{p}}^{\varphi }\left({ℝ}^n\right). Moreover, we also give sufficient conditions for the boundedness of T_{\alpha ,m,i}^b from M_{\overrightarrow{q_1}}^{{\varphi }_1}\left({ℝ}^n\right)\times ...\times M_{q_m\to }^{{\varphi }_m}\left({ℝ}^n\right)\text{to}{M}_{\overrightarrow{p}}^{\varphi }\left({ℝ}^n\right). 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.

Keywords

Generalized mixed Morrey space / multi-sublinear fractional maximal operator / multilinear fractional integral operator / BMO / commutator

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