Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma\in (0,1)$

Question

I am Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma \in (0,1)$, where $M$ is a compact manifold.

Any references are appreciated.

PS I am also looking for a reference that gives interpolation inequalities that justify (for when $u_\epsilon \in C(I, H^k) \cap C^1(I,H^{k-1})$), that $\{ u_\epsilon : \epsilon \in (0,1] \} $ is bounded in $C^{\sigma}(I,H^{k-\sigma}(M))$.

where $k$ is some nonnegative integer, $\sigma$ as above.

Thanks.

Answer 1

Surely there is a typo for the first inclusion, since you have strictly less than a derivative in $L^2(M)$ on the left hand side and one full derivative in $L^{\infty}(M)$ on the right hand side.

Regarding the interpolation question, that's an application of the usual interpolation inequality between $H^k(M)$ and $H^{k-1}(M)$ (which itself comes out from Hölder inequality). Let $t,s \in I$ be two times and $\sigma \in ]0,1[$. We have :

$$\|u(t)-u(s)\|_{H^{k-\sigma}} \lesssim \|u(t)-u(s)\|_{H^{k}}^{1-\sigma} \|u(t)-u(s)\|_{H^{k-1}}^{\sigma}. $$

Because $u \in \mathcal{C}(I,H^k)$ and $M$ is compact, the first norm of the r.h.s. is bounded. Because $u \in \mathcal{C}^1(I,H^{k-1})$, the second norm of the r.h.s. is $\lesssim |t-s|^{\sigma}$.

Thus, $$\|u(t)-u(s)\|_{H^{k-\sigma}} \lesssim |t-s|^{\sigma}$$ and you are done.