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 $f\in L^{\Phi }(X,\mu )$ to satisfy \[ \lim_{q\rightarrow \infty}\|f\|_{L^{Ψ_q}(X,μ)}=C\|f\|_{L^\infty(X,μ)}. \] 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.

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