Trace theorem in Sobolev space

Question

So I'm trying to solve a problem but I got stuck in one of the questions, and I'm sure it's related to the trace theorem, it says: Suppose $\gamma_t : D(\mathbb{R}^n) \longrightarrow D(\mathbb{R}^{n-1})$ such that $(\gamma_tu)(x')=u(x',t)$ with $x' \in \mathbb{R}^{n-1}, t \in \mathbb{R} $. Prove that for all $t_1,t_2 \in \mathbb{R}$: $$||\gamma_{t_1} - \gamma_{t_2}||^2_{L^2(\mathbb{R}^{n-1})} \leq |t_1 - t_2| ||u||_{H^1(\mathbb{R}^n)} \ \ \ \ \forall u \in D(\mathbb{R}^{n})$$ I tried to solve this using many ways but I couldn't reach anything. Any hints or starting points to solve this?

Answer 1

Do you mean $\Vert \gamma_{t_1}u-\gamma _{t_2}u\Vert_{L^2}^2$? Try to use the fundamental theorem of calculs and Cauchy–Schwarz inequality.