Extension of Submultiplicativity and Supermultiplicativity of Orlicz Functions

Abstract

Results concerning extension of submultiplicativity and supermultiplicativity for Orlicz functions are proved. A typical result is the $\text{following}$: If the Orlicz function $\varphi$ is submultiplicative at infinity, then an Orlicz function $\psi$, which is submultiplicative on ${\mathbb R}_+$, equivalent to $\varphi$ at infinity and satisfying $\psi(u)/u \to 0$ as $u \to 0$ exists if and only if the conjugate function $\varphi^*$ satisfies the $\Delta_2$-condition at infinity. Some complementary results and (counter-)examples are also included.