On the Riesz Potential and Its Commutators on Generalized Orlicz-Morrey Spaces

Abstract

We consider generalized Orlicz-Morrey spaces M-Phi,M-phi(R-n) including their weak versions WM Phi,phi(R-n). In these spaces we prove the boundedness of the Riesz potential from M-Phi,M-phi 1(R-n) to M-Psi,M-phi 2(R-n) and from M-Phi,M-phi 1(R-n) to WM Psi,phi 2(R-n). 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 (phi(1), phi(2)), which do not assume any assumption on monotonicity of phi(1)(x, r), phi(2)(x, r) in r.