Why $\int_{\mathbb R^n}\varphi(z)\int_{\mathbb R^n}\left|u(x-\frac{z}{n})\right|^p\,dx\,dz\leq \|u\|^p$

Question

Let $\varphi\in \mathcal C_0^\infty (\mathbb R^n)$ s.t. $\int_{\mathbb R^n} \varphi=1$ and $\varphi\geq 0$

Why $$\int_{\mathbb R^n}\varphi(z)\left\{\int_{\mathbb R^n}\left|u\left(x-\frac{z}{n}\right)\right|^p\,dx\right\}\,dz\leq \|u\|^p_{L^p},$$ in other word, why $$\int_{\mathbb R^n}\left|u\left(x-\frac{z}{n} \right) \right|^p \, dx\leq \int_{\mathbb R^n}|u(x)|^p \, dx.$$

Answer 1

Hint

In fact, they are equal ! Make the substitution $u=x+\frac{z}{n}$ and conclude.