We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ${ℝ}^n$ with respect to upper Ahlfors regular measures $\nu$, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure $\nu$ and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

我们研究了将任意整数阶的 Sobolev 型空间嵌入到域上的函数空间中的紧致性${ℝ}^n$关于上 Ahlfors 常规措施$\nu$，也就是说，Borel 测量其在球上的衰减受其半径的幂支配。本文考虑的 Sobolev 型空间和目标空间是建立在一般重排不变函数范数之上的。提供了几个紧实性的充分条件，并且这些条件通常也被证明是必要的，从而产生了明显的紧实性结果。值得注意的是，措施之间的唯一联系$\nu$紧凑性标准是测量在球上衰减的速度。还介绍了对基于 Lorentz-Zygmund 范数的 Sobolev 型空间的应用。