what is $L_c$ norm for the gradient?

Question

Small question, does anyone know what norm is this one?

If $f \in W^{1,p}(R^n)$ and $u \in W^{1,\infty}(R^n; R^n)$, then for $1 \leq p < \infty$

$$ ||\text{grad} f(u) ||_{L^p(R^n; R^n)} \leq \color{red}{ ||f||_{\mathcal{L}_c(W^{1,\infty}(R^n; R^n); W^{1,p}(R^n;R^n))}} ||u||_{W^{1,\infty}(R^n; R^n)}$$

Thank you in advance!

Answer 1

$$||f||_{\mathcal{L}_c(W^{1,\infty}; W^{1,p})} = \sup_{u \in W^{1,\infty}} \frac{\|f(u)\|_{W^{1,p}}}{\|u\|_{W^{1,\infty}}} $$