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 T from just one space to the full range of weighted spaces, whenever an m m -linear operator T T is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights A p → , r → A_{\vec {p}, \vec {r}} , and the limited range of the L p L^p scale. To show extrapolation theorems above, by means of a new weighted Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to build a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear ω \omega -Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.