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Suppose that $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is Lipschitz with some constant $L$ and that $F(0)=0$. Then it is clear that $F$ defines a Lipschitz continuous map $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ by $u \mapsto F(u)$ with the same Lipschitz constant $L$.

We can extend this to Sobolev spaces $H^k(\mathbb{R}^d)$ when $k \in \mathbb{N}$, by requiring in addition that $F \in C^k$ with the first $k$ derivatives of $F$ Lipschitz and that the first $k$ derivatives vanish at $0$. For instance, if $k=1$, we can compute \begin{align} \| \partial_i(F(u)-F(\tilde{u})) \|_{L^2(\mathbb{R}^d)} &= \| \nabla F(u) \partial_i u - \nabla F(\tilde{u}) \partial_i \tilde{u} \|_{L^2} \\ &\leq \| \nabla F(u) \|_{L^2} \|\partial_i u - \partial_i \tilde{u} \|_{L^2} + \|\partial_i \tilde{u} \|_{L^2} \|\nabla F(u) - \nabla F(\tilde{u}) \|_{L^2} \\ &\leq C \left( \|u\|_{L^2} \|u-\tilde{u} \|_{H^1}+\| \tilde{u} \|_{H^1} \| u- \tilde{u}\|_{L^2} \right) \\ &\leq C (\|u\|_{H^1} + \| \tilde{u} \|_{H^1}) \| u - \tilde{u} \|_{H^1} \end{align} where we have used that $\nabla F$ is again a Lipschitz map (and I denoted all constants by $C$). Thus we obtain that the map $u \mapsto F(u)$ is (at least) locally Lipschitz in $H^1$. Similarly of course for $H^k$, when $k \in \mathbb{N}$.

Now here is my question: (how) is it possible to extend this result to Sobolev spaces $H^s$ with real index $s \in \mathbb{R}$? Is the map $u \mapsto F(u)$ well-defined and locally Lipschitz continuous on the space $H^s(\mathbb{R}^d)$? Is there some interpolation argument that easily accomplishes this? Finally, did I miss something, and is the map perhaps even globally Lipschitz?

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