Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$

Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are there (good) references where these spaces are introduced and sobolev embedding theorems of these spaces can be found?

(I know how $C^{k,\alpha}(M,\mathbb{R})$, $W^k_p(M,\mathbb{R})$ and $C^{k,\alpha}(E)$, $W^k_P(E)$ for sections of a vector bundle $E\rightarrow M$ are defined if that helps.)

Also I would appreciate any good references on the topic of sobolev and hölder spaces of maps between manifolds and sections of vector bundles.

Thanks in advance!

  • $\begingroup$ For Sobolev spaces, the only way I know how to make sense of this would be to embed $N$ in some ambient Euclidean space $\mathbb{R}^N$, and then define $W^{k,p}(M,N):=\{u\in W^{k,p}(M,\mathbb{R}^N):u(x)\in N \text{ a.e.}\}$. $\endgroup$ – Matt Jun 11 '15 at 18:06

The Hölder maps are very straightforward to define in the usual way if you have a metric on both the domain and range. If $N$ is just a differentiable manifold, I guess the local version of Hölder continuity is invariant under local diffeomorphisms. So that would still be meaningful, although not the global version.

Concerning the Sobolev spaces, these are distributions in general, if my memory serves me correctly. So this is part of the general question of how to define manifold-valued distributions, which is not so easy. I think it might be impossible because for distributions, you need integration by parts of integrals of test functions. Those integrals are not well defined in general because the range space is not a vector space. So you can't compute the average of a distribution of points on the range manifold. Something similar to this question was discussed in item 935310.


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