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Note on the compactness of commutators

Tuomas Oikari

TL;DR

The paper investigates when commutators $[b,T]$ of singular integral operators are compact from $L^p$ to $L^q$ across the regimes $q>p$, $q=p$, and $q<p$. It introduces the function-space scales $X^{p,q}$ and $Y^{p,q}$ built from $\operatorname{BMO}^{\alpha}$ and $\dot{L}^r$ to formulate sufficiency conditions for compactness, and proves an elementary off-diagonal extrapolation principle that allows one to deduce diagonal compactness from off-diagonal compactness. Using Krasnosel'skii interpolation, the authors show that the known results for the three regimes (Guo–Hos, Uchiyama, Hytönen–Li–Tao–Yang) are connected and, in fact, follow from one another via the extrapolation framework. The approach provides a streamlined, unified proof strategy for compactness of commutators for both Calderón–Zygmund and rough SIOs, with an independently interesting extrapolation lemma that extends beyond the present setting. The work thus clarifies the role of symbol regularity (VMO and $\dot{L}^r$), optimizes the conceptual framework, and reinforces the interdependence of the three exponent regimes.

Abstract

The optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of singular integral operators of both Calderón-Zygmund and of rough type are shown in the different exponent ranges $``q>p"$, $``q=p"$ and $``q<p"$ to quickly follow from each other. The approach is through classical compactness interpolation methods. We also present a new elementary off-diagonal to diagonal extrapolation principle for the compactness of commutators of linear operators, which is of independent interest.

Note on the compactness of commutators

TL;DR

The paper investigates when commutators of singular integral operators are compact from to across the regimes , , and . It introduces the function-space scales and built from and to formulate sufficiency conditions for compactness, and proves an elementary off-diagonal extrapolation principle that allows one to deduce diagonal compactness from off-diagonal compactness. Using Krasnosel'skii interpolation, the authors show that the known results for the three regimes (Guo–Hos, Uchiyama, Hytönen–Li–Tao–Yang) are connected and, in fact, follow from one another via the extrapolation framework. The approach provides a streamlined, unified proof strategy for compactness of commutators for both Calderón–Zygmund and rough SIOs, with an independently interesting extrapolation lemma that extends beyond the present setting. The work thus clarifies the role of symbol regularity (VMO and ), optimizes the conceptual framework, and reinforces the interdependence of the three exponent regimes.

Abstract

The optimal sufficient conditions for the -to- compactness of commutators of singular integral operators of both Calderón-Zygmund and of rough type are shown in the different exponent ranges , and to quickly follow from each other. The approach is through classical compactness interpolation methods. We also present a new elementary off-diagonal to diagonal extrapolation principle for the compactness of commutators of linear operators, which is of independent interest.
Paper Structure (3 sections, 5 theorems, 19 equations)

This paper contains 3 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. An elementary compactness extrapolation principle for commutators
  3. Applications

Key Result

Theorem 1.1

Let $p_i,q_i \in (1,\infty)$ and $T$ be a linear operator that is $L^{p_0}(\mathbb{R}^n)\to L^{q_0}(\mathbb{R}^n)$ compact and $L^{p_1}(\mathbb{R}^n)\to L^{q_1}(\mathbb{R}^n)$ bounded. Let $\theta\in (0,1)$ be arbitrary and set $1/p := \theta/p_1 + (1-\theta)/p_2$ and $1/q := \theta/q_0 + (1-\theta)

Theorems & Definitions (10)

  • Theorem 1.1: Kras60
  • Lemma 1.5: Off-diagonal to diagonal compactness extrapolation principle for commutators
  • proof
  • Theorem 2.1: Guo et al. GHWY21; Hytönen et al. HOS2023
  • Theorem 2.2: Uchiyama Uch1978
  • Theorem 2.3: Hytönen et al. HLTY2023
  • proof : Proof of Theorem \ref{['thm:Uchiyama']} assuming Theorem \ref{['thm:GuoHos']}
  • proof : Proof of Theorem \ref{['thm:HLTY']} assuming Theorem \ref{['thm:Uchiyama']}
  • proof : Proof of \ref{['eq:Chain2']}, i.e. of Theorems \ref{['thm:GuoHos']} and \ref{['thm:Uchiyama']} assuming Theorem \ref{['thm:HLTY']} or \ref{['thm:Uchiyama']}
  • proof : Proof of \ref{['eq:Chain3']}, i.e. of Theorems \ref{['thm:Uchiyama']} and \ref{['thm:HLTY']} assuming Theorem \ref{['thm:Uchiyama']} or \ref{['thm:GuoHos']}