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 \(r\).