Lack of interpolation of linear operators in spaces of smooth functions

We prove that $C^k(\Omega)$, the space of $k$ times continuously differentiable functions on the closure of a region in a finite-dimensional manifold, is not an interpolation space between $C(\Omega)$ and $C^n(\Omega)$ for $0<k<n$. We find analogous results for the Sobolev–Stein spaces. In the class of spaces $C_\varphi$, defined by the modulus of continuity, we describe all interpolation spaces between $C$ and $C^2$.
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