Families of Young Functions and Limits of Orlicz Norms

Given a $\sigma$-finite measure space $(X,\mu )$, a Young function $\mathrm{\Phi }$, and a one-parameter family of Young functions $\{{\mathrm{\Psi }}_q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^{\mathrm{\Phi }}(X,\mu )$ to satisfy $$\underset{q\to \mathrm{\infty }}{\lim }\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{{\mathrm{\Psi }}_q}(X,\mu )}=C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(X,\mu )}\mathrm{.}$$ The constant $C$ is independent of $f$ and depends only on the family $\{{\mathrm{\Psi }}_q\}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.