Families of Young Functions and Limits of Orlicz Norms

Scott Rodney; Sullivan F. MacDonald

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arXiv · 2209.01149

Families of Young Functions and Limits of Orlicz Norms

Scott Rodney·Sullivan F. MacDonald

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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.

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Scott Rodney, Sullivan F. MacDonald. 2022-09-02. Families of Young Functions and Limits of Orlicz Norms. https://arxiv.org/abs/2209.01149

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