@PedroLauridsenRibeiro Thanks for the comments. (1) I see your points. To be honest, I didn't think about that. But in this case, the question concerns the "standard" definition using weak derivatives. (2) I was not aware of this, but it makes sense of course. In my understanding, when taking about the Sobolev embedding theorem, I want a continuous embedding, hence w.r.t. the global $C^{l}$-norm.

@AlpinistKitten see the appendum of my question above.

@IgorKhavkine Thanks a lot for your comment! Thats indeed interesting, but unfortunately, not quite what I am looking for. I am a bit suprised that there seems to be no general result. At least for the Poisson equation on complete manifold, I think I have seen the claim somewhere that there is a Green's function $G:C^{\infty}_{c}(M)\to L^{2}(M)$, but I might be wrong though.

If $\alpha$ is a k-form with $k>0$, then the equation $d^{\ast}d\alpha=0$ is not elliptic. Instead, $(d^{\ast}d+dd^{\ast})\alpha=0$ is.