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Extrapolation of compactness on weighted mixed Lebesgue spaces

María J. Carro, Carlos Pérez, Rodolfo H. Torres

TL;DR

This work extends compactness methods to weighted mixed Lebesgue spaces $L^p_uL^q_v$ by establishing a Fréchet-Kolmogorov type compactness criterion and a Rubio de Francia–style uniform compact extrapolation framework. The authors show that uniform compactness on a base space $L^{p_0}(w)$ transfers to all $L^p_uL^q_v$ with $u\in A_p$, $v\in A_q$, enabling robust extrapolation of compactness properties. They apply the theory to commutators with $CMO$ and to pseudodifferential operators, obtaining uniform compactness results across the weighted mixed-scale setting. The results rely on $A_p$ weight theory, sharp maximal functions, and extrapolation techniques, providing a systematic method to transfer compactness across anisotropic weighted spaces with potential impact on harmonic analysis and PDEs.

Abstract

A version of the Fréchet-Kolmogorov theorem for the compactness of operators in weighted mixed Lebesgue spaces is proved and a corresponding compact extrapolation theory a la Rubio de Francia is developed. Several applications are presented too.

Extrapolation of compactness on weighted mixed Lebesgue spaces

TL;DR

This work extends compactness methods to weighted mixed Lebesgue spaces by establishing a Fréchet-Kolmogorov type compactness criterion and a Rubio de Francia–style uniform compact extrapolation framework. The authors show that uniform compactness on a base space transfers to all with , , enabling robust extrapolation of compactness properties. They apply the theory to commutators with and to pseudodifferential operators, obtaining uniform compactness results across the weighted mixed-scale setting. The results rely on weight theory, sharp maximal functions, and extrapolation techniques, providing a systematic method to transfer compactness across anisotropic weighted spaces with potential impact on harmonic analysis and PDEs.

Abstract

A version of the Fréchet-Kolmogorov theorem for the compactness of operators in weighted mixed Lebesgue spaces is proved and a corresponding compact extrapolation theory a la Rubio de Francia is developed. Several applications are presented too.

Paper Structure

This paper contains 9 sections, 13 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weights
  4. Sharp maximal operators
  5. Extrapolation
  6. Compactness in weighted mixed Lebesgue spaces
  7. Applications
  8. Commutators of singular integrals and point-wise multiplication
  9. Pseudodifferential operators

Key Result

Theorem 2.1

Let $\mathcal{F}$ be a family of pairs of functions $(f, g)$ such that, for some $1\leq p_0<\infty$, every $w\in A_{p_0}$, every $(f,g)\in\mathcal{F}$, and an increasing function $\varphi$, Then, for every $1<p<\infty$ there exists and increasing function $\phi_p$ depending only $\varphi$ such that for every $w\in A_{p}$ and every $(f,g)\in\mathcal{F}$,

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Corollary 2.6
  • proof
  • Theorem 3.1
  • ...and 10 more