Interpolation theory and $C^k$-spaces

Question

Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that makes it easier). I have a few questions regarding their interpolation theory:

• Is $C^1(M)$ an interpolation space for the pair $(C^0(M),C^2(M))$? According to Bergh-Lofström, this means that any linear map $T:C^0(M)\rightarrow C^0(M)$ which leaves $C^2(M)$ invariant, also leaves $C^1(M)$ invariant. I don't see how one would prove this. The reason I am wondering is that this would be a sufficient (but not a necessary) condition for the association $(C^0,C^2)\mapsto C^1$ to extend to an interpolation functor on Banach spaces (Aronszajn-Gagliardo Theorem).
• Can we identify the interpolation spaces $[C^k, C^l]_\theta$ or $[C^k,C^l]_{\theta,p}$ (where the brackets stand for complex and real interpolation respectively)? I only find results of this kind for Hölder-Zygmund spaces $C_*^k$, which differ from $C^k$ for integer values of $k$. Maybe one can even identify $C^k$ as member of some larger scale of spaces (Besov, Triebel, etc.)?

Answer 1

$C^1$ is not an interpolation space between $C$ and $C^2$. There is an example due to Mitjagin and Semenov of a sequence $T_n$ of uniformly bounded operators both in $C,C^2$ whose norm blow up in $C^1$. You find this example in the booklet by A. Lunardi "Interpolation Theory" (SNS Pisa publisher). In the edition in my hands, it is in Chapther I, after Example 1.3.3, pag 29.