Extrapolation of compactness on weighted spaces

Abstract

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 $w\in A_p({\mathbb{R}}^d)$. This “compact version” of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of 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. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one (“off-diagonal estimates”) or only in a limited range of the $L^p$ scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner–Riesz multipliers.