Sobolev embedding in the space of continuous functions [duplicate]

Question

Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be the space of continuous function from $I$ to $\mathbb{R}$.

Question: How can one prove that there exists an inclusion $W^{1,2}(I,\mathbb{R}) \hookrightarrow C^{0}(I,\mathbb{R})$ which is bounded (continuous)?

In the case where $I$ is a open bounded interval the above is definitely true (by proving it first for smooth function with compact support and then using a density arguement). What about the case when $I$ is unbounded?

Greetings, Dani

PS: In order to apply the idea of Pietro Majer from the post When is Sobolev space a subset of the continuous functions?, isn't it necessary $\Omega \subset \mathbb{R}$ to be bounded subset? I do not understand his arguement in the case $\Omega = \mathbb{R}$ ?.

Answer 1

Your question is not clear. If you want an embedding into the space of continuous bounded functions on R, then all you need is an estimate on each bounded subinterval, which you say you already understand. So what else do you need?

To give a complete answer: by density, it is sufficient to prove an estimate for a $C^1$ function $u(x)$, in which case for all $x,x_0$ belonging to an interval $I$ of length 1 you can write $$ |u(x)|\le|u(x_0)|+|\int_{x_0}^xu'(t)dt|\le |u(x_0)|+\|u'\|_{L^2(I)} $$ and taking as $x_0$ the point where $|u|$ reaches its minimum on $\overline{I}$ we can continue $$ \le\|u\|_{L^2(I)}+\|u'\|_{L^2(I)}\lesssim\|u\|_{W^{1,2}}. $$ This gives $\|u\|_{L^\infty}\lesssim\|u\|_{W^{1,2}}$ on $R$