Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$.
Do we have that $(u_n)$ is bounded in $W^{1, p}$?
2026-04-13 08:22:21.1776068541
Question about sequence being bounded in $W^{1, p}$?
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Yes. More precisely, $\|u_n\|_{1,p}=\|\varphi\|_{1,p}$ for every $n$. In every integral in the $W^{1,p}$ norm, you simply need to make the change of variable $y=x+n$, whose Jacobian is always 1.
Edit: graphically, $u_n$ is simply a left shift of $\varphi$, and horizontal shifts do not change Lebesgue integrals (over $\mathbb{R}^n$).