I am studying trace theorems in unbounded domains, but I do not find any results about what is the situation in unbounded domains.
I am interested in the following domain: $$ (-\infty, a)\cup(b,+\infty), $$ where $a\neq b$. There exist trace theorems for this kinds of domains ?
Since the boundary of the domain is just 2 points, there's no concern with "integrability" of the function on the boundary.
Instead, the only concern is whether there is a well-defined boundary value of your function. And this follow from Morrey-Sobolev (or since we are in one-dimension: fundamental theorem of calculus), which states that $W^{k,p} \hookrightarrow C^0$ (in the one dimensional setting) for all $k,p \geq 1$. In this case the continuous representative can be extended continuously to the boundary, and hence the boundary value is well-defined.
(I expect for reasonable definitions of fractional Sobolev spaces you can extend this also to $W^{s,p}$ with $s > 1/p$ and $p > 1$.)