Generalized Riesz potential operators on Musielak–Orlicz–Morrey spaces over unbounded metric measure spaces

In this paper we discuss the boundedness of the Hardy–Littlewood maximal operator \(M_{\lambda }, \ \lambda \ge 1\), and the variable Riesz potential operator \(I_{\alpha (\cdot ),\tau }, \ \tau \ge 1\), on Musielak–Orlicz–Morrey spaces \(L^{\Phi ,\kappa ,\theta }(X)\) over unbounded metric measure spaces X. As an important example, we obtain the boundedness of \(M_{\lambda }\) and \(I_{\alpha (\cdot ),\tau }\) in the framework of double phase functionals with variable exponents \(\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}, \ x \in X, \ t \ge 0\), where \(p(x)<q(x)\) for \(x\in X\), \(a(\cdot )\) is a non-negative, bounded and Hölder continuous function of order \(\theta \in (0,1]\). Our results are new even for the variable exponent Morrey spaces or for the doubling metric measure case in that the underlying spaces need not be bounded.