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I wanna understand a proof of the Sobolev-Gagliardo-Nirenberg inequality. Therefore, I need to know why $|f|^q \in C_c^1(\mathbb{R}^n)$ for $f \in C_c^1(\mathbb{R}^n)$ and $q>1$. Can eventually someone tell me why this holds?

Matthis Stresemann
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1 Answers1

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The function $\varphi(x) = |x|^q$ is continuously differentiable on $\mathbf R$ for all $q > 1$ and you can calculate its derivative explicitly: $$\varphi'(x) = \left\{ \begin{array}{ll} qx |x|^{q-2} & x \not= 0 \\ 0 & x=0 \end{array} \right.$$

Now use the chain rule.

Umberto P.
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