What is the domain of momentum operator on $mathbb{R}$?

Observables in QM are postulated to be self-adjoint operators. Those have to obey $hat A vphantom{A}^+ ! = hat A$, including the equality of their domains. If we work on a finite interval $(a, b)$, an example of such an observable is the momentum operator:
$$
hat p: {rm D}(hat p) to L^2 big( (a,b) big) [5pt]
{rm D}(hat p) = big{; f in W^{1,2} big( (a,b) big) ; big| ; f(a+) = f(b-) ;big} [5pt]
hat p f = -{rm i} f’
$$

We can easily inspect that $hat p$ with this domain is indeed self-adjoint using integration by parts:
$$
big( hat p f, ; g big)_{L^2}
= {rm i} big( f’, ; g big)_{L^2}
= big[ fg big]_a^b – {rm i} big( f, ; g’ big)_{L^2}
= big[ fg big]_a^b + big( f, ; -{rm i}g’ big)_{L^2}
$$
Here, $g$ has to be from $W^{1,2}$ in order to have a derivative and the necessary and sufficient condition for $[fg]_a^b$ to be zero is $g(a+) = g(b-)$, hence ${rm D}(hat p^+) = {rm D}(hat p)$ and $hat p$ is self-adjoint.

However, this doesn’t work on infinite intervals. In $L^2(mathbb{R})$, functions either don’t have a limit at infity, or it’s zero. If we require that $f(-infty) to 0, ;; f(+infty) to 0$, it is sufficient for $g$ to be only bounded at infinity and we get ${rm D}(hat p^+) subsetneq {rm D}(hat p)$. On the other hand, if we require that $f$ is bounded at infinity, we get that $g$ has to vanish at infinity, therefore ${rm D}(hat p^+) !supsetneq {rm D}(hat p)$.

How do I achieve ${rm D}(hat p^+) = {rm D}(hat p)$ on $L^2(mathbb{R})$? What is the domain of the momentum operator on $mathbb{R}$?