Weak compactness criteria in $L_1$ space in terms of Orlicz function

TL;DR

This paper establishes equivalence criteria for weak compactness in $L_1$ spaces using Orlicz functions, providing direct proofs and extending results to $L_1(0, \infty)$.

Contribution

It offers a direct proof of the equivalence of two classical criteria for weak compactness in $L_1$ spaces via Orlicz functions, including the case on $L_1(0, \infty)$.

Findings

Proves equivalence of Chong's and De la Vallée Poussin's criteria using Orlicz functions.

Extends the equivalence results to $L_1(0, \infty)$.

Provides a direct proof approach for these criteria.

Abstract

In this paper, we provide a direct proof for the equivalence of K.M. Chong's and De la Vall\'{e}e Poussin's criteria of weak compactness of a subset $K$ of $L_1(0,1)$ in terms of some Orlicz function. Furthermore, we discuss the equivalence in $L_1(0, \infty)$.