Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

Abstract

In this paper, we establish the necessary and sufficient conditions for the boundedness of fractional maximal operator $$M_{\alpha }$$ and the fractional maximal commutators $$M_{b,\alpha }$$ in Orlicz $$L^{\Phi }(X)$$ and generalized Orlicz–Morrey spaces $$\mathcal {M}^{\Phi ,\varphi }(X)$$ on spaces of homogeneous type $$X=(X,d,\mu )$$ in the sense of Coifman-Weiss.