Whitney embedding theorem for Hölder manifolds

Question

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?

Answer 1

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 $C^{r,\alpha}$.