Families of Young Functions and Limits of Orlicz Norms

Abstract: Given a $\sigma$-finite measure space $(X,\mu)$, a Young function $\Phi$, and a one-parameter family of Young functions ${\Psi_q}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L<sup>\Phi(X,\mu)$ to satisfy [ \lim_{q\rightarrow \infty}|f|{L^{\Psi_q}(X,\mu)}=C|f|{L^\infty(X,\mu)}. ] The constant $C$ is independent of $f$ and depends only on the family ${\Psi_q}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.