Let \(\mathcal{Y}\) be the set of all Young functions \(\Phi\) such that, for any \(t\in(0,\infty)\), \(\Phi(t)\in(0,\infty)\). Assume that \(\mathcal{G}_1\) is the set of all functions \(\phi:\;[0,\infty)\to[0,\infty)\) such that \(\phi(t)\) is nondecreasing but \(\phi(t)/t\) is nonincreasing. Moreover, for any \(\Phi\in\mathcal{Y}\), let \(\mathcal{G}_2\) be the set of all functions \(\phi:\;[0,\infty)\to[0,\infty)\) such that \(\phi(t)\) is nondecreasing, but for any \(s\in(0,\infty)\), \(\phi((s+t)^n)/\Phi^{-1}(((s+t)/s)^n)\) is nonincreasing, where \(\Phi^{-1}\) denotes the inverse of \(\Phi\).
Let \(\Phi\in\mathcal{Y}\), \(\varphi\in\mathcal{G}_1\) and \(\phi\in\mathcal{G}_2\). Denote by \(\mathcal{Q}\) the family of all cubes in \(\mathbb{R}^n\) with sides parallel to the coordinate axes. For the cube \(Q\in\mathcal{Q}\) and the measurable function \(f\) on \(\mathbb{R}^n\), let \[ \|f\|_{(\varphi,\Phi);Q}:=\inf\left\{\lambda\in(0,\infty): \frac{\varphi(|Q|)}{|Q|}\int_Q \Phi\left(\frac{|f(x)|}{\lambda}\right)\,dx\leq1\right\}. \] Then the generalized Orlicz-Morrey space \(\mathcal{L}^{\varphi,\Phi}(\mathbb{R}^n)\) is defined to be the Banach space with the norm \[ \|f\|_{\mathcal{L}^{\varphi,\Phi}(\mathbb{R}^n)}:=\sup_{Q\in\mathcal{Q}}\|f\|_{(\varphi,\Phi);Q}. \] Furthermore, for \(Q\in\mathcal{Q}\) and the measurable function \(f\) on \(\mathbb{R}^n\), let \[ \|f\|_{\Phi;Q}:=\inf\left\{\lambda\in(0,\infty): \frac{1}{|Q|}\int_Q \Phi\left(\frac{|f(x)|}{\lambda}\right)\,dx\leq1\right\}. \] Then the generalized Orlicz-Morrey space \(\mathcal{M}_{\phi,\Phi}(\mathbb{R}^n)\) is defined to be the Banach space with the norm \[ \|f\|_{\mathcal{M}_{\phi,\Phi}(\mathbb{R}^n)}:=\sup_{Q\in\mathcal{Q}}\phi(|Q|)\|f\|_{\Phi;Q}. \]
In this article, the authors investigate the differences between the Orlicz-Morrey spaces \(\mathcal{L}^{\varphi,\Phi}(\mathbb{R}^n)\) and \(\mathcal{M}_{\phi,\Phi}(\mathbb{R}^n)\) in some typical cases.