Extrapolation for multilinear compact operators and applications

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.