Consider $u \in W^{1,2}(\mathbb{R}_{+})$. Is there always a bounded representative of $u$?
I have come across this sort of question multiple times, but I just cannot wrap my head around it. This is connected, at least from my point of view and from my experience, to this more generic question: given a function $u \in L^p(\mathbb{R})$, for $p \neq \infty$, under which additional hypothesis (if any) is $u$ bounded?
I have often found myself with integrability hypotheses over unbounded sets such as $\mathbb{R}$ and have hoped to deduce some boundedness, but I can't think clearly about it.
We have the continuity of the embedding $W^{1,2}(0,1)$ into $C([0,1])$. That is there is $c>0$ such that for all $u$ $$ \|u\|_{L^\infty(0,1)} \le c \|u\|_{W^{1,2}(0,1)} . $$ This is true for all intervals of length $1$, so we can sum this up: $$ \sum_{n=-\infty}^\infty \|u\|_{L^\infty(n,n+1)}^2 \le c^2 \sum_{n=-\infty}^\infty \|u\|_{W^{1,2}(n,n+1)}^2 = c^2 \|u\|_{W^{1,2}(\mathbb R)}^2. $$ Hence, the series on the left is converging, in particular $\|u\|_{L^\infty(\mathbb R)} \le c \|u\|_{W^{1,2}(\mathbb R)}$. Moreover, the series tells us that $$ \lim_{|n|\to\infty}\|u\|_{L^\infty(n,n+1)}=0, $$ i.e., $u$ kind of vanishes at infinity.