论文标题
Sobolev嵌入和规范衰减的紧凑性
Compactness of Sobolev embeddings and decay of norms
论文作者
论文摘要
我们研究了基于重排的空间构建的sobolev空间嵌入的紧凑性之间的关系，并在对球衰减的速度的一定限制下，赋予了$ d $ -ahlfors措施。我们表明，这种嵌入的紧凑性的门户虽然可以通过最佳嵌入和几乎紧凑的嵌入方式正式描述，但却是非常难以捉摸的。众所周知，当其目标空间具有最佳基本函数时，这种Sobolev嵌入并不紧凑。我们表明，令人惊讶的是，这样的目标空间实际上可以“从根本上扩大”，但所产生的嵌入仍然没有2级。为此，我们开发了两种不同的方法。一个是基于扩大最佳目标空间本身的基础，另一个基于扩大与最佳基本函数相对应的Marcinkiewicz空间。
We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed of its decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space can actually be "fundamentally enlarged", and yet the resulting embedding remains noncompact. In order to do that, we develop two different approaches. One is based on enlarging the optimal target space itself, and the other is based on enlarging the Marcinkiewicz space corresponding to the optimal fundamental function.
