Extrapolation of compactness on variable $L^{p(\cdot)}$ spaces

Abstract: Building on a recent approach of Hyt\"onen-Lappas to the extrapolation of compactness of linear operators on weighted $L^p(w)$ spaces, we extend these results to the weighted variable-exponent spaces $L^{p(\cdot)}(w)$. Related results are recently due to Lorist-Nieraeth, who showed that compactness can be extrapolated from $L^p(w)$ to a general class of Banach function spaces including the $L^{p(\cdot)}(w)$ spaces. The novelty of our result is that one can take any variable-exponent $L^{p(\cdot)}(w)$, not just $L^p(w)$, as a starting point of extrapolation. An application of our extrapolation to commutators $[b,T]$ of pointwise multipliers and singular integrals allows us to complete a set of implications, showing that $b\in CMO(\mathbb{R}^d)$ is not only sufficient (as known from Lorist-Nieraeth) but also necessary for the compactness of $[b,T]$ on any fixed $L^{p(\cdot)}(w)$.