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I was wondering about the following fact: if $\Omega$ is a bounded subset of $\mathbb{R}^n$ and $u\in W^{1,p}(\Omega)$ and $g\in C^1(\mathbb{R},\mathbb{R})$ such that $|g'(t)t|+|g(t)|\leq M$, is it true that $g\circ u \in W^{1,p}(\Omega)$?

If $g'\in L^{\infty}$ this would be true, but here we don't have this kind of estimate...

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up vote 1 down vote accepted

You get $g' \in L^\infty$ from the assumptions, since $|g'(t)| \le M/|t|$, and $g'$ is continuous at $0$.

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Of course, thank you! –  balestrav Oct 16 '12 at 15:29
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