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Suppose $\Omega\subset\mathbb R^n$ is a bounded open set. Suppose $kp>n$ (where $10$ such that we have $||f\circ u||_{W^{k,p}}\le C||f||_{C^k}(1+||u||_{L^\infty}^{k-1})(1+||u||_{W^{k,p}})$ for all $u\in C^\infty_0(\Omega)$ and $f\in C^k(\mathbb R)$.

For $k=1$, the proposition seems to be true even without the assumption $p>n$. Is this correct? Also, how do I go about proving the statement for $k>1$?

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    You need to use the chain rule and the Sobolev embeddings. Write down the partial derivatives of $f$ and use Sobolev embeddings to bound them in terms of the right hand side. – 2017-02-05
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    And regarding $p \leq n$, this is a bit delicate. Note that is is not clear that if $p \leq n$, the function $f \circ u$ even belongs to $L^p$. If I remember correctly, this holds but it is not as trivial as the case $p > n$. – 2017-02-05
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    @levap: Can you please supply some more details? I'm still unable to see how $||u||_{L^\infty}$ shows up in the estimate. – 2017-02-05

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