Explain why $\int_{\mathbb R^N}|\nabla u_r(x)|^p dx=r^p \int_{\mathbb{R}^N}|\nabla u(rx)|^p dx$

Question

Suppose $$ (*) \quad \|u\|_{L^q(\mathbb{R^N})} \le c \|\nabla u\|_{L^q(\mathbb{R^N;\mathbb{R^N}})}. $$ Assume $u\in C_c^1(\mathbb{R}^N)$ and for $r>0$ defined the rescaled function $$u_r(x) := u(rx), \quad x \in \mathbb{R}^N.$$

If $(*)$ holds for $u_r$ then \begin{align} \bigg(\int_{\mathbb{R}^N}|u(rx)|^q dx\bigg)^{1/q} & = \bigg(\int_{\mathbb{R}^N}|u_r(x)|^q dx\bigg)^{1/q} \\ & \le c\bigg(\int_{\mathbb{R}^N}|\nabla u_r(x)|^p dx\bigg)^{1/p} \\ & = c\bigg(r^p \int_{\mathbb{R}^N}|\nabla u(rx)|^p dx\bigg)^{1/p} \\ \end{align}

I don't see where the $r^p$ term is appearing from?

Answer 1

Hint: By Chain rule we have $\partial_i [u(rx) ]= r[\partial_i u](rx) .$ This yields

$$[\nabla u_r](x) =[\nabla u(\cdot r)](x)=r [\nabla u](rx)$$ Then integrates on both side.