# Invariance under diffeomorphisms of the Sobolev $H^s$ spaces

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?

• I guess the result should be obvious once you write down the definition for $H^s_{loc}(\Omega)$. But if you insist on a reference, I would recommend Friedlander's Introduction to the theory of distributions. – Hui Yu Nov 11 '12 at 3:32
• Not for me, and the reference you provide doesn't help at all... – levap Nov 12 '12 at 17:10
• Really? Can you tell me what did you get and where are you stuck? – Hui Yu Nov 12 '12 at 17:13
• Say I work with the Fourier transform definition. So $u \in H^s(\Omega)$ if for all $\phi \in C^{\infty}_c(\Omega)$, we have $u\phi \in H^s(\mathbb{R}^n)$, where $H^s(\mathbb{R}^n)$ is defined using Fourier transform conditions. I have no idea how to estimate the Fourier transform of the composition $u \circ \psi$ of $u$ with a diffeomorphism in terms of the Fourier transform of $u$. – levap Nov 12 '12 at 17:38
• I think it's better to consider $H^s$ as the Besov space $B^s_{2,2}$ and use their local characterization. The invariance under diffeomorphisms ought to be someone in the 1st or 2nd volume of Triebel's Function Spaces but this is only a guess, and I don't have access to the book now. – user53153 Dec 22 '12 at 2:40

## 1 Answer

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 $k$.