Extrapolation of compactness on weighted spaces: Bilinear operators

TL;DR

This paper extends weighted extrapolation techniques to bilinear operators, enabling the transfer of compactness properties across weighted spaces and improving results on bilinear Calderón-Zygmund operators and related areas.

Contribution

It develops a new approach to extrapolate compactness for bilinear operators using softer tools, avoiding complex criteria, and enhances existing weighted compactness results.

Findings

Recovered and improved weighted compactness results for bilinear operators

Developed a new extrapolation method avoiding Fréchet–Kolmogorov criterion

Extended compactness results to broader classes of bilinear operators

Abstract

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method.