Inclusion properties of Orlicz and weak Orlicz spaces

TL;DR

This paper investigates the inclusion properties of Orlicz and weak Orlicz spaces on Euclidean space, providing necessary and sufficient conditions for when one space is contained within another, based on their structural characteristics.

Contribution

It introduces new criteria for inclusion relations in Orlicz and weak Orlicz spaces, advancing understanding of their structure and norm computations.

Findings

Derived necessary and sufficient conditions for inclusion

Computed norms of characteristic functions of Euclidean balls

Enhanced understanding of the structure of Orlicz spaces

Abstract

In this paper we discuss the structure of Orlicz spaces and weak Orlicz spaces on $\mathbb{R}^n$. We obtain some necessary and sufficient conditions for the inclusion property of these spaces. One of the keys is to compute the norm of the characteristic functions of the balls in $\mathbb{R}^n$.