On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces

We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\Rn)$ including their weak versions $WM_{Φ,φ}(\Rn)$. In these spaces we prove the boundedness of the Riesz potential from $M_{Φ,φ_1}(\Rn)$ to $M_{Ψ,φ_2}(\Rn)$ and from $M_{Φ,φ_1}(\Rn)$ to $WM_{Ψ,φ_2}(\Rn)$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(φ_{1},φ_{2})$, which do not assume any assumption on monotonicity of $φ_{1}(x,r)$, $φ_{2}(x,r)$ in r.

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