论文标题
在sobolev空间的紧凑型子集上
On compact subsets of Sobolev spaces on manifolds
论文作者
论文摘要
通常，在$ \ mathbb {r}^m $上定义的sobolev空间具有非紧密的嵌入到$ l^p $ - 空间中，但是它具有该子空间，该子空间为此嵌入了紧凑。有三种众所周知的这样的子空间的情况，即雷利希紧凑性，用于在有界域上的函数子空间（或无限的域，无限的无限域），strauss compactness，用于$ \ mathbb {r}^m $ and proge ebolev sobolev sobess $ \ mathbb {r}^m $ s s s s s s s s s s space s raduss cartacts space。 Strauss紧凑性的已知概括包括具有块状对称性的功能子空间，在Riemannian歧管上具有某些对称性的功能子空间，以及更一般的BESOV和Triebel-Lizorkin空间的类似子空间。可以用对应于商空间的有效尺寸较小的临界Sobolev指数的上升来解释对称性的存在。
It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $\mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of functions with certain symmetries on Riemannian manifolds, as well as similar subspaces of more general Besov and Triebel-Lizorkin spaces. Presence of symmetries can be interpreted in terms of the rising critical Sobolev exponent corresponding to the smaller effective dimension of the quotient space.
