紧流形上的Sobolev-Slobodeckij空间
Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited
摘要：
在这篇文章中，我们提出了一个连贯的严格概述的主要性质的Sobolev-Slobodeckij空间的部分，向量丛在紧凑的流形，这种类型的结果是分散在文献中，很难找到。特别强调了具有非整数光滑阶的空间，并特别注意到对于Rn中的一般非光滑区域U，0<t<1，1<p<oo，W（1，p）（U）不一定连续嵌入W（t，p）（U）中这一特殊事实.这对Sobolev-Slobodeckij空间的乘法性质以及随后对流形上的索伯列夫空间的研究产生了可怕的后果。据作者所知，一些证明，特别是那些有关的Sobolev-Slobodeckij空间的一般向量丛部分的性质，不能在文献中找到的一般性出现在这里。
英文摘要：
In this article we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain U in Rn, 0<t<1, and 1<p<oo, it is not necessarily true that W(1,p)(U) is continuously embedded in W(t,p)(U). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. To the authors' knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.
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