Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

Abstract

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.

107 pages. Version 1 was extensive monograph; it was recommended we split into three papers. First paper maintains title/theme of original, now version 2 (arXiv:1704.07930v2). Second paper is current version, focusing on Sobolev-Slobodeckij spaces on compact manifolds. Third paper (arXiv:1806.02188) is expansion of Appendix F in original paper, focusing on Locally Sobolev-Slobodeckij spaces