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This paper is devoted to studying the Rubio de Francia extrapolation formultilinear 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 isindeed established in terms of the multilinear Muckenhoupt weights $A_{\vec{p},\vec{r}}$, and the limited range of the $L^p$ scale. To show extrapolationtheorems above, by means of a new weighted Fr\'{e}chet-Kolmogorov theorem, wepresent the weighted interpolation for multilinear compact operators. To provethe latter, we also need to bulid a weighted interpolation theorem inmixed-norm Lebesgue spaces. As applications, we obtain the weighted compactnessof 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. Beyondthat, we establish the weighted compactness of higher order Calder\'{o}ncommutators, and commutators of Riesz transforms related to Schr\"{o}dingeroperators.

Extrapolation for multilinear compact operators and applications