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A Marcinkiewicz-Zygmund inequality and the Kadec Pełczynśki theorem in Orlicz spaces

Istvan Berkes, Eduard Stefanescu, Robert Tichy

Abstract

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pełczyński-type result -- originally established by the first and third authors for $L^p$ spaces with $1 \le p < 2$ -- to a broader class of Orlicz spaces defined via Young functions $ψ$ satisfying $x \le ψ(x) \le x^2$.

A Marcinkiewicz-Zygmund inequality and the Kadec Pełczynśki theorem in Orlicz spaces

Abstract

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pełczyński-type result -- originally established by the first and third authors for spaces with -- to a broader class of Orlicz spaces defined via Young functions satisfying .

Paper Structure

This paper contains 6 sections, 5 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Main Results
  4. Proof of Khinchin's Inequality \ref{['Khinch']} and \ref{['khi2']}
  5. Proof of the Marcinkiewicz-Zygmund inequality \ref{['mz']}
  6. Proof of the generalized Kadec Pełczyński theorem \ref{['kp']}

Key Result

Lemma 3.1

Let $r_n$ be the $n$-th Rademacher function. Let $\psi$ be a Young-function with $(\cdot)\ll_\psi\psi\ll_\psi e^{(\cdot)}$ and let $x_1,\ldots,x_N\in \mathbb{C}$. Then

Theorems & Definitions (13)

  • Lemma 3.1: Khinchin's inequality in Orlicz spaces
  • Theorem 3.2: Marcinkiewicz–Zygmund inequality in Orlicz spaces
  • Remark
  • Remark
  • Lemma 3.3: Khinchin's inequality in Lorentz spaces
  • Theorem 3.4: Marcinkiewicz–Zygmund inequality in Lorentz spaces
  • Remark
  • Theorem 3.5: Generalized Kadec-Pełczyński theorem
  • proof : Proof of Lemma \ref{['khi']}
  • proof : Proof of Lemma \ref{['khi2']}
  • ...and 3 more