论文标题
与实值函数空间相关的映射组和Sobolev-lie组的直接限制
Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups
论文作者
论文摘要
让$ m $成为尺寸$ m $（无边界）的紧凑平滑分歧，而$ g $是有限维的谎言组，lie代数$ g $。令$ h^{> m/2}（m，g）$是所有映射$γ\ colon m \ to g $的组，对于某些$ s> m/2 $，为$ h^s $。我们表明，$ h^{> m/2}（m，g）$可以在米尔诺的意义上成为常规的谎言组，在silva空间上建模$ h^{> m/2} $，这是希尔伯特space $ h^s（m，g）$ s> m/2 $ s> m/2 $的局部凸出直接限制，例如$ s> m/2 $作为平滑的谎言组，Hilbert-lie Groups $ h^s（m，g）$。我们还解释了$ h^s（m，g）$上的（已知的）谎言组结构是如何作为一种特殊情况，即每当开放子集的$ f（u，r）$ f（u，r）$ f（u，r）$ f（u，r）$ u $ u $ u $ r^m $时，给出了简单的axioms，请给出简单的$ r^m $。
Let $M$ be a compact smooth manifold of dimension $m$ (without boundary) and $G$ be a finite-dimensional Lie group, with Lie algebra $g$. Let $H^{>m/2}(M,G)$ be the group of all mappings $γ\colon M\to G$ which are $H^s$ for some $s>m/2$. We show that $H^{>m/2}(M,G)$ can be made a regular Lie group in Milnor's sense, modelled on the Silva space $H^{>m/2}(M,g)$ which is the locally convex direct limit of the Hilbert spaces $H^s(M,g)$ for $s>m/2$, such that $H^{>m/2}(M,G)$ is the direct limit of the Hilbert-Lie groups $H^s(M,G)$ for $s>m/2$ as a smooth Lie group. We also explain how the (known) Lie group structure on $H^s(M,G)$ can be obtained as a special case of a general construction of Lie groups $F(M,G)$ whenever real-valued function spaces $F(U,R)$ on open subsets $U$ of $R^m$ are given, subject to simple axioms.
