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Let $\Omega = S^1 \times [a, b] $ and $f\in C^\infty (\Omega )$. When is $\;H^s(\Omega, \Omega) \rightarrow H^s(\Omega, \Omega), u \mapsto f \circ u$ well defined and smooth?

I've only found results on open domains or closed manifolds, but nothing for closed, noncompact sets.

Edit: and what about $\Omega=\mathbb{R} \times [a, b] $?

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