Extrapolation for multilinear compact operators and applications

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. 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.