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Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces

Nuno J. Alves

Abstract

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic $L_p$ spaces on $\mathbb{R}^n$. These are nonlocally convex $\mathrm{F}$-spaces that contain the standard $L_p$ spaces as dense subspaces and include all measurable functions supported on sets of finite measure. In contrast with the classical $L_p$ setting, an additional almost equiboundedness condition is needed, and we prove that together with the natural tail and translation conditions it characterizes relative compactness. We conclude with illustrative examples.

Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces

Abstract

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic spaces on . These are nonlocally convex -spaces that contain the standard spaces as dense subspaces and include all measurable functions supported on sets of finite measure. In contrast with the classical setting, an additional almost equiboundedness condition is needed, and we prove that together with the natural tail and translation conditions it characterizes relative compactness. We conclude with illustrative examples.

Paper Structure

This paper contains 6 sections, 5 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Asymptotic Lp spaces
  3. Main result
  4. Proof of Theorem \ref{['main_thm']}: Necessity
  5. Proof of Theorem \ref{['main_thm']}: Sufficiency
  6. Examples

Key Result

Theorem 1.1

A subset $\mathcal{F} \subseteq L^p(\mathbb{R}^n)$, with $1 \leq p < \infty$, is totally bounded with respect to $\| \!\cdot\! \|_p$ if and only if the following two conditions hold:

Theorems & Definitions (13)

  • Theorem 1.1: Kolmogorov--Riesz compactness theorem in $L^p(\mathbb{R}^n)$ hanche2010kolmogorovhanche2019improvement
  • Theorem 3.1: Kolmogorov--Riesz compactness theorem in $\Lambda^p(\mathbb{R}^n)$
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Example 6.1
  • Example 6.2
  • ...and 3 more