Families of Young Functions and Limits of Orlicz Norms

TL;DR

This paper characterizes when a family of Orlicz norms converges to the essential supremum norm, providing necessary and sufficient conditions and illustrating with examples and counterexamples.

Contribution

It establishes precise conditions for the convergence of Orlicz norms to the supremum norm as the parameter tends to infinity.

Findings

Identifies necessary and sufficient conditions for norm convergence.

Provides examples satisfying the conditions.

Includes counterexamples where conditions fail.

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 \[ \lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \] 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.