Soblolev space question

Question

I want to prove that if $u(x)$ is in $H^s(\mathbb R^n)$ then $u(cx+d)$ is also in $H^s(\mathbb R^n)$ for any constant $c$ and $d$ .Can someone help me?Thanks

Answer 1

This doesn't hold for all values, I.e. take $c=0$ then your function is constant and thus is not in $L^2$. Assume $c\ne 0$ then the map you have is an affine change of variables, so change variables back in the integral to deduce the result (note that this affine map maps from $\mathbb R^n$ to $\mathbb R^n$).