It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces \(L^{p,\infty }\) has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as \(p<\infty \). Finally, we show that if the target space is \(L^{\infty }\) (which formally is also a weak Lebesgue space with \(p=\infty \)), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.

众所周知，到弱Lebesgue空间的尖锐Sobolev嵌入不是紧致的，但是这种嵌入的不紧致程度是否等于其算子范数的问题构成了一个众所周知的开放问题。现有理论提出了这样一个论点，即如果目标范数是不相交的超可加性，则可能会解决该问题，但是空间\（L ^ {p，\ infty} \）的不相交的超可加性问题也已经开放。在本文中，我们解决了这两个问题。我们首先显示弱的Lebesgue空间永远不会相交地具有超加性，因此排除了建议的技术。但是随后我们表明，也许有些令人惊讶的是，锐利的Sobolev嵌入的非紧致程度的度量与嵌入规范仍然一致，至少在\（p <\ infty \）。最后，我们证明了，如果目标空间是\（L ^ {\ infty} \）（它也是形式为\（p = \ infty \）的弱Lebesgue空间），那么事物本质上是不同的。为了给出包括这种情况在内的全面答案，我们基于一种相当意外的组合论证开发了一种新方法，并由此证明了一般原理，其特殊情况意味着在这种情况下，非紧凑性的度量严格小于它的规范。我们开发了一种技术，使我们能够准确评估这种非紧凑性。