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

Yerlan Nessipbayev, Kanat Tulenov

Abstract

In this paper, we provide a direct proof for the equivalence of K.M. Chong's and De la Vallée 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 $L_1(0, \infty)$.

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

Abstract

In this paper, we provide a direct proof for the equivalence of K.M. Chong's and De la Vallée Poussin's criteria of weak compactness of a subset of in terms of some Orlicz function. Furthermore, we discuss the equivalence in .

Paper Structure

This paper contains 8 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Marcinkiewicz spaces
  4. Orlicz spaces
  5. Equivalence of Chong's and De la Vallée Poussin's criteria in $L_1(0,1)$
  6. Equivalence of Chong's and De la Vallée Poussin's criteria in $L_1(0,\infty)$
  7. Appendix
  8. Acknowledgment

Key Result

Lemma 3.1

For any integrable function $f$ on $I=(0, \infty)$, there exists an $N$-function $G$ such that $G(|f|)$ is integrable on $I$.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • Theorem 3.5
  • proof
  • ...and 6 more