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Compactness of averaging operators on non-reflexive Lebesgue spaces

Katsuhisa Koshino

Abstract

Let $X$ be a Borel and Borel-regular metric measure space whose closed balls are of positive and finite measure. In this paper, we shall give equivalent conditions for averaging operators on non-reflexive Lebesgue spaces $L^1(X)$ and $L^\infty(X)$ on X to be compact, where X has some doubling property and satisfies certain uniform continuity between metric and measure.

Compactness of averaging operators on non-reflexive Lebesgue spaces

Abstract

Let be a Borel and Borel-regular metric measure space whose closed balls are of positive and finite measure. In this paper, we shall give equivalent conditions for averaging operators on non-reflexive Lebesgue spaces and on X to be compact, where X has some doubling property and satisfies certain uniform continuity between metric and measure.
Paper Structure (6 sections, 19 theorems, 43 equations)

This paper contains 6 sections, 19 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Averaging operators on Banach function spaces
  3. Compact subsets in $L^p(X)$
  4. Compactness of averaging operators on $L^1(X)$ and $L^\infty(X)$
  5. The case of $L^1(X)$
  6. The case of $L^\infty(X)$

Key Result

Theorem 1.1

Let $X$ be a metric measure space having the $s$-doubling property and the property $(\star)_s$ for any $s > 0$. Then $A_r : L^1(X) \to L^1(X)$ is compact if and only if $X$ is bounded.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Theorem 1.1 of Kos25
  • Theorem 2.2: Theorem 1.3 of Kos25
  • Proposition 2.3
  • Theorem 2.4
  • Corollary 2.5: Corollary 1.4 of Kos25
  • Theorem 3.1
  • Proposition 3.2
  • Lemma 3.3
  • ...and 19 more