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Compactness of averaging operators on Banach function spaces

Katsuhisa Koshino

Abstract

Let $X$ be a Borel metric measure space such that each closed ball is of positive and finite measure. In this paper, we give a sufficient and necessary condition for averaging operators on a Banach function space $E(X)$ on $X$ to be compact. As a corollary, we show that the averaging operators on the Lorentz space $L^{p,q}(X)$ of $X$ is compact if and only if $X$ is bounded, in the case where $X$ is a doubling and Borel-regular metric measure space with some continuity between metric and measure.

Compactness of averaging operators on Banach function spaces

Abstract

Let be a Borel metric measure space such that each closed ball is of positive and finite measure. In this paper, we give a sufficient and necessary condition for averaging operators on a Banach function space on to be compact. As a corollary, we show that the averaging operators on the Lorentz space of is compact if and only if is bounded, in the case where is a doubling and Borel-regular metric measure space with some continuity between metric and measure.
Paper Structure (3 sections, 11 theorems, 46 equations)

This paper contains 3 sections, 11 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['suff.']} and \ref{['nec.']}
  3. Proof of Corollary \ref{['equiv.lorentz']}

Key Result

Theorem 1.1

Suppose that $A_r(E(X)) \subset E(X)$ and the following conditions hold: If $\mu(X) < \infty$, then $A_r : E(X) \to E(X)$ is compact.

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['suff.']}
  • proof : Proof of Theorem \ref{['nec.']}
  • ...and 12 more