I am looking for a reference regarding the higher (than the first) order Hölder spaces on Riemannian manifolds. I am aware that defining Hölder spaces of form $C^{0,\alpha}$ is not an issue even between two metric spaces. However, so far, I haven't even seen a definition of higher order Hölder spaces outside of the Euclidean setting.
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$\begingroup$ Are you asking about the spaces $C ^{k, r}$ with $k \in \mathbb N$ and $r \in (0,1)$? If so, they are defined in all the books that prove the Sobolev embedding theorem on Riemannian manifolds (Hebey, Aubin, Grigor'yan and many others). $\endgroup$– Alex M.Commented Jun 17 at 19:37
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$\begingroup$ @AlexM Yes, I would like a definition of spaces $C^{k,r}$, preferably with a norm. (Also, the Riemannian manifold need not be compact). I'm not sure if that is correct for all the books you've mentioned, but I think Aubin and Hebey might define them for compact manifolds, or at least use maps from atlas to make sense of subtracting vectors from different tangent spaces (without, for example, requiring that for every pair of points we have unique minimal distance path connecting them). I would like to know if there is a definition that does not use compactness, maps, or uniqueness of paths. $\endgroup$– Kacper KurowskiCommented Jun 17 at 23:15
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2 Answers
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Not a full blown answer, but still it may help a bit:
I developed a bit of this in my PhD thesis that was published in a book series, see also my website for freely available thesis and book preprint. The contents is in appendix C and specifically used in Lemma C.10 to estimate the $C^{k,\alpha}$ norm of a flow on a Riemmanian manifold (of bounded geometry), given that the first derivative of it satisfies an exponential growth estimate. I didn't define spaces of $C^{k,\alpha}$ functions really, as I didn't define proper $C^{k,\alpha}$ norms, see the discussion around the lemma.
I'm not aware of other (earlier) results in the literature, but to be honest I haven't searched exhaustively for it.
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In the 1980ies Hans Triebel wrote a number of papers and a textbook on function spaces of all kinds. One of these is his 1987 Paper in the Mathematische Nachrichten, Characterizations of Function Spaces on a Complete RIEMANNian Manifold with Bounded Geometry (capitalization by the author).
In this paper he looks at the Besov spaces $B^s_{p,q}(M)$. Holder spaces are the special case $p=q=\infty$. It should be possible to present this in a simpler way if you only want the case of Holder spaces, but I am not aware of a reference for that.
Triebel's paper is here:
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1$\begingroup$ Minor correction: according to MathSciNet, Triebel has been publishing since 1962 while his latest reviewed paper on MathSciNet is from 2023. So in writing "in the 1980ies..." I seriously understated Triebel's contributions $\endgroup$ Commented Apr 25 at 15:10