I want to prove that for a function $F\in C^k(\mathbb{R}^n)$ which vanishes at zero, and a function $u\in H^k(\mathbb{R}^n)$we get:
$$\left\| \int_{r=0}^1 F'(ru)(\cdot)dr \right\|_{L^{\infty}} \leq \sup_{|v| \leq c\|u\|_{H^k}} |F'(v)(\cdot)|$$ where $c$ is some constant. So far I got the following:
$$\left\| \int_{r=0}^1F'(ru)(\cdot)dr \right\|_{L^{\infty}} \leq \sup_{x\in \mathbb{R}^n}\sup_{r\in [0,1]} |F'(ru)(x)|$$
I don't see how to arrive at this bound: $\sup_{|v|\leq c\|u\|_{H^k}}|F'(v)(\cdot)|$.
Any advice or hints are welcomed. PS $k$ is an integer.