Families of Young Functions and Limits of Orlicz Norms

This paper provides conditions on a Young function $\Phi$ and a one-parameter family of Young functions $\{\Psi_q\}$ that ensure the Orlicz norms associated with each $\Psi_q$ have the following property for any $f\in L^\Phi(X,\mu)$, \[ %c\|f\|_{L^\infty(\Omega)}\le \lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \] The constant $C$ is independent of $f$ and depends only on the family $\{\Psi_q\}$. Moreover, our conditions are necessary and sufficient if $f\in L^1(X,\mu)$. Several examples of one-parameter families satisfying our conditions are given, along with counterexamples when our conditions fails. Applications in partial differential equations are also discussed