Compact sets in quasi-Banach function spaces and multilinear extrapolation of compactness on $L^{p(\cdot)}(w)$

This paper addresses a novel Riesz--Kolmogorov theorem on quasi-Banach function spaces under mild conditions and the extrapolation of multilinear compact operators in the context of weighted variable Lebesgue spaces. We establish the latter result via our Riesz--Kolmogorov theorem which yields a wei