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Extrapolation for bilinear compact operators in the variable exponent setting

Spyridon Kakaroumpas, Stefanos Lappas

TL;DR

The paper develops a comprehensive extrapolation framework for compactness of bilinear operators on weighted variable-exponent Lebesgue spaces. It combines an abstract diagonal/off-diagonal extrapolation principle with the Cobos–Fernández-Cabrera–Martínez interpolation theorem, supported by a factorization theory for multilinear variable-exponent weights and Lions–Peetre interpolation. The authors then apply these methods to derive new compactness results for commutators of bilinear ω-Calderón–Zygmund operators, bilinear fractional integrals, and bilinear Fourier multipliers in the variable-exponent weighted setting, unifying prior works. This advances the understanding of compactness in the multilinear, variable-exponent, weighted landscape and expands the toolkit for proving refined regularity properties in harmonic analysis.

Abstract

We establish extrapolation of compactness for bilinear operators in the scale of weighted variable exponent Lebesgue spaces. First, we prove an abstract principle relying on the Cobos-Fernández-Cabrera-Martínez theorem. Then, as an application we deduce new compactness results for the commutators of bilinear $ω$-Calderón-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers acting on weighted variable exponent Lebesgue spaces. Our work extends and unifies among others earlier works of the second named author together with Hytönen as well as Oikari.

Extrapolation for bilinear compact operators in the variable exponent setting

TL;DR

The paper develops a comprehensive extrapolation framework for compactness of bilinear operators on weighted variable-exponent Lebesgue spaces. It combines an abstract diagonal/off-diagonal extrapolation principle with the Cobos–Fernández-Cabrera–Martínez interpolation theorem, supported by a factorization theory for multilinear variable-exponent weights and Lions–Peetre interpolation. The authors then apply these methods to derive new compactness results for commutators of bilinear ω-Calderón–Zygmund operators, bilinear fractional integrals, and bilinear Fourier multipliers in the variable-exponent weighted setting, unifying prior works. This advances the understanding of compactness in the multilinear, variable-exponent, weighted landscape and expands the toolkit for proving refined regularity properties in harmonic analysis.

Abstract

We establish extrapolation of compactness for bilinear operators in the scale of weighted variable exponent Lebesgue spaces. First, we prove an abstract principle relying on the Cobos-Fernández-Cabrera-Martínez theorem. Then, as an application we deduce new compactness results for the commutators of bilinear -Calderón-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers acting on weighted variable exponent Lebesgue spaces. Our work extends and unifies among others earlier works of the second named author together with Hytönen as well as Oikari.

Paper Structure

This paper contains 20 sections, 30 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variable Lebesgue spaces
  4. Notation for variable exponent functions
  5. Luxemburg norm of variable Lebesgue spaces
  6. Weighted variable Lebesgue spaces
  7. Log-Hölder continuity
  8. Variable exponent Muckenhoupt characteristics
  9. Auxiliary results
  10. Factorization results for multilinear Muckenhoupt weights with variable exponents
  11. Lions--Peetre compactness theorem in the bilinear setting
  12. Banach function spaces
  13. The Cobos--Fernández-Cabrera--Martínez theorem
  14. Proof of abstract main results
  15. Applications
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $T$ be a bilinear operator such that there exist fixed constants $t\in(0,\infty)$ and $\gamma\in\left[0,\infty\right)$ satisfying both of the following two assumptions. Then: For all proper $2$-admissible quadruples $(\vec{p}(\cdot), q(\cdot),\vec{1},\infty)$ such that and and for all $2$-tuples of weights $\vec{w}$ with $\vec{w}\in\mathcal{A}_{\vec{p}(\cdot),q(\cdot)}$ and $\vec{w}^{t}\in\

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.6
  • Remark 2.8
  • ...and 59 more