Exotic $C^k$ manifolds

Question

How much is known about exotic $C^k$-manifolds? For example,

1. Is it known whether there are $C^k$-differentiable manifolds that are homeomorphic, but not $C^k$ diffeomorphic?
2. More generally, is it known whether there are $C^k$-differentiable manifolds that are $C^l$ diffeomorphic, but not $C^k$ diffeomorphic for $l < k$?
3. Do any of the famous examples for $k=\infty$ (such as $\mathbb R^4$ or Milnor-Kervaire's exotic spheres) give examples for $k < \infty$?

Answer 1

There are no such exotic manifolds. Whitney proved in

Whitney, Hassler Differentiable manifolds. Ann. of Math. (2) 37 (1936), no. 3, 645–680.

that every $C^1$-differential structure can be uniquely smoothed to a $C^{\infty}$ differential structure. This was improved by Morrey and Grauert to show that in fact it can be uniquely smoothed to a real analytic structure; see

H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958), 460-472.

C. B. Morrey, The analytic embedding of abstract real-analytic manifolds, Ann. of Math. 68 (1958), 159-201.