Extrapolation of compactness on weighted spaces

Tuomas Hytönen

Published 2020-03-03Version 1

Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ for all $\tilde w\in A_{p_1}(\mathbb R^d)$ and in addition compact on $L^{p_1}(w_1)$ for some $w_1\in A_{p_1}(\mathbb R^d)$. Then $T$ is bounded and compact on $L^p(w)$ for all $p\in(1,\infty)$ and all $w\in A_p$. This ``compact version'' of Rubio de Francia's weighted extrapolation theorem follows from a combination of classical results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. As quick corollaries, we recover some recent results on the compactness of Calder\'on--Zygmund singular integral operators and their commutators on weighted spaces.