Extrapolation of compactness on weighted spaces

Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ all $\tilde w\in A_{p_1}(\mathbb R^d)$ and in addition compact $L^{p_1}(w_1)$ $w_1\in R^d)$. Then $L^p(w)$ $p\in(1,\infty)$ $w\in A_p(\mathbb This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from combination results the interpolation theory spaces one hand, operators abstract other hand. Moreover, generalizations this compactness are obtained that space to different ("off-diagonal estimates") or only limited range $L^p$ scale. As applications, we easily recover several recent commutators singular integral operators, fractional integrals pseudo-differential obtain new about Bochner-Riesz multipliers.