Multilinear extrapolation of compactness on mixed-norm spaces

In this paper, we develop the Rubio de Francia extrapolation theorem for the multilinear compactness on mixed-norm Lebesgue spaces. More precisely, if a multilinear operator is bounded on weighted product spaces, then its compactness can be extrapolated from unweighted product spaces to the full range of weighted mixed-norm spaces. This result is mainly based on a multilinear interpolation theorem of compactness on weighted mixed-norm spaces, for which we present a characterization of compactness on mixed-norm spaces and a multilinear interpolation theorem of boundedness on multi-mixed-norm spaces. As applications of the extrapolation theorem, we obtain compactness results for several kinds of bi-parameter operators on weighted mixed-norm spaces, including multilinear bi-parameter Calder\'{o}n-Zygmund operators, multilinear bi-parameter dyadic paraproducts, bilinear bi-parameter continuous paraproducts, and bilinear bi-parameter pseudo-differential operators.