A note on the limit of Orlicz norms

TL;DR

This paper extends a classical inequality relating $L^p$ norms to the $L^ty$ norm by generalizing it to Orlicz spaces, broadening the understanding of norm limits in functional analysis.

Contribution

It introduces a generalization of the limit behavior of $L^p$ norms to Orlicz norms, expanding the theoretical framework of norm convergence.

Findings

Established the limit of Orlicz norms as a generalization of the $L^p$ to $L^ty$ case.

Provided a mathematical proof extending classical inequalities to Orlicz spaces.

Enhanced the theoretical understanding of norm limits in functional analysis.

Abstract

We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $p\rightarrow\infty$ is the $L^\infty$ norm to the scale of Orlicz spaces.