TITLE: Invariance under diffeomorphisms of the Sobolev $H^s$ spaces
QUESTION [5 upvotes]: Let $\Omega, \Omega' \subset \mathbb{R}^n$ be two open subsets and let $\psi : \Omega \rightarrow \Omega'$ be a $C^k$ diffeomorphism. Then, $\psi$ induces by pullback a linear isomorphism 
$$u \mapsto u \circ \psi$$
between the Sobolev spaces $W^{k,p}_{\text{loc}}(\Omega')$ and $W^{k,p}_{\text{loc}}(\Omega)$. I assume that there is an analogous result for the Hilbert-Sobolev spaces $H^s_{\text{loc}}(\Omega)$ with real exponent $s$. Can someone please provide me a reference where this is discussed?

REPLY [4 votes]: Use the theory of operator interpolation (see for instance, the book Interpolation Spaces by Jöran Bergh, Jörgen Löfström).
The idea is as follows.  Let $\Psi$ be the linear operator defined by $\Psi u = u \circ \psi$.  Prove the following:
$$ \Psi \in \mathcal{L}(H^k(\Omega),H^k(\Omega)) \text{ and } \Psi \in \mathcal{L}(H^{k-1}(\Omega),H^{k-1}(\Omega))$$
by using the chain rule and change of variables.  Then interpolation theory provides the estimate
$$ \Psi \in \mathcal{L}(H^s(\Omega),H^s(\Omega)) \text{ for } s \in (k-1,k)$$
since $H^s$ is the interpolation between $H^{k-1}$ and $H^{k}$.  Moreover, we get an estimate
$$ \lVert \Psi \rVert_{H^s \to H^s} \le C \lVert \Psi \rVert_{H^{k-1} \to H^{k-1}}^\theta \lVert \Psi \rVert_{H^k \to H^k}^{1-\theta} $$
where $\theta \in (0,1)$ is determined by $s$ and $k$.