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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 \(r\).

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