Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ for all $\tilde w\in A_{p_1}$ and in addition compact on $L^{p_1}(w_1)$ for some $w_1\in A_{p_1}$. 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 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. As quick corollaries, we recover several recent results on the compactness of singular integral operators and their commutators on weighted spaces.

中文翻译：

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加权空间上紧致性的外推

令 $T$ 是一个线性算子，对于某些 $p_1\in(1,\infty)$，对于所有 $\tilde w\in A_{p_1 }$ 并在 $L^{p_1}(w_1)$ 上紧缩一些 $w_1\in A_{p_1}$。然后对于所有 $p\in(1,\infty)$ 和所有 $w\in A_p$，$T$ 在 $L^p(w)$ 上是有界和紧致的。Rubio de Francia 著名的加权外推定理的这种“紧凑版本”一方面来自加权空间的插值和外推理论的结果，另一方面来自抽象空间上的紧凑算子的结果。作为快速推论，我们恢复了最近关于奇异积分算子及其交换子在加权空间上的紧致性的几个结果。