Decay of Sobolev functions in one dimension

Question

Suppose $u\in H^3(\mathbb{R})$. I know that since we are in one dimensional setting, $u$ is continuous (i.e., has continuous representative). What can we say about the decay of $u$ and its derivatives against "x", namely can we say anything about limits such as $\displaystyle\lim_{|x|\rightarrow\infty}xu(x)$ or $\displaystyle\lim_{|x|\rightarrow\infty}x|u'(x)|^2?$

Answer 1

This is only a partial, giving an idea how to disprove such statements (or find limits for such claims).

Let $\psi\in C_c([0,1])$ be non-zero, $\epsilon\in (0,\frac12)$. Define $$ u(x):= \sum_{n=1}^\infty n^{-1/2+\epsilon}\psi(x-n). $$ Then $$ \|u\|_{H^k}^2 = \sum_{n=1}^\infty n^{-1+2\epsilon} \|\psi\|_{H^k}^2 <\infty $$ for all $k$. On the interval $[n,n+1]$, we have $$ |u(x)| \le n^{-1/2+\epsilon} \|\psi\|_{L^\infty(\Omega)} \le (x-1)^{-1/2+\epsilon} \|\psi\|_{L^\infty(\Omega)} $$ implying $$ x^{1/2-2\epsilon} |u(x)| \to0 $$ for $|x|\to\infty$, similarly we get $$ x^{1-4\epsilon}|u'(x)|^2 \to 0. $$