$$(\Phi ,\Psi )$$ ( Φ , Ψ ) -admissible potential operators and their commutators on vanishing Orlicz-Morrey spaces

Abstract

We study the boundedness of $$(\Phi ,\Psi )$$ -admissible potential operators and their commutators on vanishing generalized Orlicz-Morrey spaces $$VM_{\Phi ,\varphi }(\mathbb {R}^n)$$ including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Riesz potential, fractional maximal operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities involving the Young functions $$\Phi $$ , $$\Psi $$ and the function $$\varphi (x,r)$$ defining the space, without assuming any monotonicity property of $$\varphi (x,r)$$ on $$r$$ .