# Composition of Sobolev map and smooth function.

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 $$\Omega=\mathbb{R} \times [a, b]$$?