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arXiv · 1704.07930

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

A. Behzadan·M. Holst
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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.

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A. Behzadan, M. Holst. 2017-04-25. Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited. https://arxiv.org/abs/1704.07930

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