# Whitney embedding theorem for Hölder manifolds

According to a result of Whitney any $$C^r$$-manifold, $$r\geq 1$$, is $$C^r$$-homeomorphic to a smooth embedded submanifold of some Euclidean space; see Theorem 1 in

Hassler Whitney, "Differentiable manifolds", Annals of Mathematics (2) 37 (1936), no. 3, pp. 645–680, JFM 62.1454.01, MR1503303, Zbl 0015.32001.

I expect that an analogous result should be true for $$C^{r,\alpha}$$-Hölder manifolds, $$r\geq 1$$, $$\alpha\in[0,1]$$, and I am wondering if there exists a reference for such a statement. Does perhaps anybody know such a reference?

Actually, I am not so much interested in the embedding, but in a refinement of the $$C^{r,\alpha}$$-Hölder structure to a smooth structure, i.e. in a $$C^{r,\alpha}$$-Hölder homeomorphism to a smooth manifold.

In particular, I am interested in the case in which the manifold to start with is itself a ($$C^{r,\alpha}$$-)embedded submanifold of some Euclidean space and in the case $$\alpha=1$$. Maybe it is easier to say something here?

Every $$C^1$$ manifold admits a compatible $$C^\infty$$ structure. You can find a proof in Hirsch's "Differential topology". It is actually quite easy and based on a fact that smoothing a $$C^1$$ diffeomorphism by convolution leads to a smooth diffeomorphism (because derivatives converge uniformly) on a slightly smaller domain.
In particular $$C^{r,\alpha}$$ manifolds, $$r\geq 1$$, admit a compatible $$C^\infty$$-structure and hence such a manifold can be embedded as a smooth submanifold of an Euclidean space. However, the embedding will only be $$C^{r,\alpha}$$ smooth since the original manifold is only $$C^{r,\alpha}$$.
• The $C^{\infty}$ structure might be compatible with the $C^1$ structure, but why should it be compatible with the $C^{r,\alpha}$ structure? If it is not, the resulting embedding is not $C^{r,\alpha}$ smooth with respect to the $C^{r,\alpha}$ structure we started with, only with respect to a distinct $C^{r,\alpha}$ structure (that is induced by the $C^{\infty}$ structure). Oct 2, 2022 at 17:54
• I have the impression that one needs e.g. a Hölder analogue of Theorem 2.7 in Hirsch's book: "Let $M$ and $N$ be $C^s$ manifolds $1\leq s \leq \infty$. Then $\mathrm{Diff}^s(M,N)$ is dense in $\mathrm{Diff}^r(M,N)$ in the strong $C^r$ topology, $1\leq r <s$." The proof of this theorem using convolutions seems to go through also in the Hölder case. Thanks for the reference. Oct 3, 2022 at 8:01