Extrapolation for bilinear compact operators in the variable exponent setting

TL;DR

This paper develops an extrapolation method for compact bilinear operators in weighted variable exponent Lebesgue spaces, unifying and extending previous results in the field.

Contribution

It introduces an abstract extrapolation principle for compactness and applies it to various bilinear operators in variable exponent spaces, advancing the theoretical framework.

Findings

Established an abstract extrapolation principle for compactness.

Derived new compactness results for bilinear Calderón-Zygmund operators.

Unified previous results and extended them to variable exponent settings.

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-Fern\'{a}ndez-Cabrera-Mart\'{i}nez theorem. Then, as an application we deduce new compactness results for the commutators of bilinear $\omega$-Calder\'{o}n-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 Hyt\"{o}nen as well as Oikari.