The properties of four elements in Orlicz-Musielak spaces (Q2770417)

Let \(X_\rho\) be a modular space ordered by a pointed convex cone \(K\) and let \(D\subset X_\rho\) be a lattically convex subset. The modular \(\rho\) is said to satisfy the upper property of four elements (UPFE) with respect to \(D\) and \(K_{\mathcal E}\) if for any \(x,y\in X_\rho\) such that \(x\geq y\) and for any \(w,z\in D\) we have NEWLINE\[NEWLINE\rho(x-w)+ \rho(y-z)\leq \rho(x-w\wedge z)+ \rho(x-w\vee z).NEWLINE\]NEWLINE It is proved that if \(\Phi\) is a finite, convex \(\varphi\)-function for all \(t\in\Omega\), \(D\) is a lattically convex subset of \(X_{\rho \Phi}\) and \(K=\{x\in X_{\rho\Phi} :x\geq 0\}\), then \(\rho\Phi\) has the property (UPFE) with respect to \(D\) and \(K\). The property (UPFE) is applied to show the existence of antiprojection operators \(P^a_D\) for which there exist \(w\in P_D^a (x)\) and \(v\in P_D^a(y)\) such that \(v\geq w\). Also, the reverse properties of four elements in Orlicz-Musielak spaces are studied.