标题： 可变空间$L^{p(\cdot)}$上的紧性外推 显示英文标题

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

摘要： 基于 Hytönen-Lappas 最近关于权加权 $L^p(w)$ 空间上线性算子紧性的外插方法，我们将这些结果推广到了权可变指数空间 $L^{p(\cdot)}(w)$。相关结果最近由 Lorist-Nieraeth 得出，他们证明了紧性可以从 $L^p(w)$ 外插到包括 $L^{p(\cdot)}(w)$ 空间在内的广义 Banach 函数空间类。我们结果的新颖之处在于可以选择任意的可变指数 $L^{p(\cdot)}(w)$ 作为外插的起点，而不仅仅是 $L^p(w)$。 我们将我们的外推法应用于点乘子和奇异积分的交换子$[b,T]$，从而完成了一组蕴含关系，表明$b\in CMO(\mathbb{R}^d)$不仅如 Lorist-Nieraeth 所知的那样是充分的，而且对于$[b,T]$在任意固定的$L^{p(\cdot)}(w)$上的紧性也是必要的。 显示英文摘要

摘要： 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 $L^{p(\cdot)}(w)$.