Boundedness of Fractional Maximal Operator and its Commutators on Generalized Orlicz–Morrey Spaces

Abstract

We consider generalized Orlicz–Morrey spaces $M_{\Phi ,\varphi }({\mathbb {R}^n})$ including their weak versions $WM_{\Phi ,\varphi }({\mathbb {R}^n})$. We find the sufficient conditions on the pairs $(\varphi _{1},\varphi _{2})$ and $(\Phi , \Psi )$ which ensures the boundedness of the fractional maximal operator $M_{\alpha }$ from $M_{\Phi ,\varphi _1}({\mathbb {R}^n})$ to $M_{\Psi ,\varphi _2}({\mathbb {R}^n})$ and from $M_{\Phi ,\varphi _1}({\mathbb {R}^n})$ to $WM_{\Psi ,\varphi _2}({\mathbb {R}^n})$. As applications of those results, the boundedness of the commutators of the fractional maximal operator $M_{b,\alpha }$ with $b \in BMO({\mathbb {R}^n})$ on the spaces $M_{\Phi ,\varphi }({\mathbb {R}^n})$ is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights $\varphi (x,r)$, which do not assume any assumption on monotonicity of $\varphi (x,r)$ on $r$.