Extrapolation for multilinear compact operators and applications

TL;DR

This paper develops a new extrapolation theory for multilinear compact operators using weighted inequalities and applies it to various multilinear operators and commutators, establishing their weighted compactness.

Contribution

It introduces a novel extrapolation framework for multilinear compact operators based on weighted inequalities and develops new weighted interpolation theorems.

Findings

Weighted extrapolation of compactness from one space to all weighted spaces.

Weighted compactness of commutators of multilinear operators.

Extension of compactness results to higher order Calderón and Riesz transform commutators.

Abstract

This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of $T$ from just one space to the full range of weighted spaces, whenever an $m$-linear operator $T$ is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights $A_{\vec{p}, \vec{r}}$, and the limited range of the $L^p$ scale. To show extrapolation theorems above, by means of a new weighted Fr\'{e}chet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to bulid a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear $\omega$-Calder\'{o}n-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calder\'{o}n commutators, and commutators of Riesz transforms related to Schr\"{o}dinger operators.