# Criterion for the coincidence of strong and weak Orlicz spaces

## TL;DR

This paper establishes precise conditions under which strong and weak Orlicz spaces are equivalent, revealing that this occurs mainly in exponential spaces and providing the exact embedding constants.

## Contribution

It offers necessary and sufficient criteria for the equivalence of strong and weak Orlicz spaces, including the exact embedding constants, a novel characterization in functional analysis.

## Key findings

- Coincidence occurs mainly in exponential spaces.
- Exact embedding constants are determined.
- Provides necessary and sufficient conditions for space equivalence.

## Abstract

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We find also the exact value of the embedding constant which appears in the corresponding norm inequality.