I asked this [question][1] on Stack Exchange previously. 
This seems to be very basic property of Sobolev spaces, but I wasn't able to find a proof for it.

> For any $s \leq 1/2$,
$$H^s(\mathbb{R}) = H^s_-(\mathbb{R}) + H^s_+(\mathbb{R})$$
where $H^s_-(\mathbb{R}) = \{ u \in H^s(\mathbb{R}) \mid \operatorname{supp} u \subset (-\infty, 0] \}$ and $H^s_+(\mathbb{R}) = \{ u \in H^s(\mathbb{R}) \mid \operatorname{supp} u \subset [0, \infty) \}$.

I am only really interested in the case $s < 1/2$, but I am led to believe the result also holds for $s = 1/2$.

**Context**

This question stems from my attempt to prove subjectivity of the natural map
$$H^s_+(\mathbb{R}) \rightarrow H^s([0, \infty)) = H^s(\mathbb{R}) / H^s_-(\mathbb{R})$$
and its higher dimensional analogues $H^s_+(\mathbb{R} \times \mathbb{R}^d) \rightarrow H^s([0, \infty) \times \mathbb{R}^d)$.
This property is claimed in Melrose's [notes][2], Proposition 3.5.1.
The proof involves taking the dual of the injection 
$H^{-s}_+(\mathbb{R}) \rightarrow H^{-s}([0,\infty))$.
But this requires the natural topology of $H^{-s}_+(\mathbb{R})$ to coincide with the subspace topology from $H^{-s}([0,\infty))$, which is not justified in the notes. 
I believe the latter property is equivalent to $H^{-s}_-(\mathbb{R}) + H^{-s}_+(\mathbb{R})$ being closed in $H^{-s}(\mathbb{R})$.

**My thoughts**

In the case $s < 1/2$, I can show that $H^s_-(\mathbb{R}) + H^s_+(\mathbb{R})$ contains the Schwartz functions $\mathcal{S}(\mathbb{R})$, but this does not imply the desired result unless $H^s_-(\mathbb{R}) + H^s_+(\mathbb{R})$ is closed.

The inclusion $\mathcal{S}(\mathbb{R}) \subset H^s_-(\mathbb{R}) + H^s_+(\mathbb{R})$ is obtained by multiplying $u \in \mathcal{S}(\mathbb{R})$ by sequences of smooth functions approximating the indicators on $(-\infty, 0]$ and $[0, \infty)$, and showing that the resulting sequences converge in $H^s$. 
For example, fixing a smooth $\phi$ supported on $[0, \infty)$ and eventually equal to 1, then consider its scalings $\phi_{\epsilon}(x) = \phi(x /\epsilon)$.
One can show that $\lim_{\epsilon \rightarrow 0^+} u \phi_{\epsilon}$ converges in $H^s$, $s \in [0, 1/2)$ using the inequality $\|v\|_{H^s} \lesssim \|v\|_{H^0}^{1-s} \|v\|_{H^1}^s$ and the crude estimates
$$\|u(\phi_\epsilon - \phi_\delta)\|_{L^2} \leq \|u\|_{L^{\infty}} \|\phi_\epsilon - \phi_\delta\|_{L^\infty} \|\phi_\epsilon - \phi_\delta\|_{L^0}^{1/2} \lesssim \epsilon^{1/2}$$
$$\|u \cdot \partial(\phi_\epsilon - \phi_\delta)\|_{L^2} \leq \|u\|_{L^\infty} \|\partial(\phi_\epsilon - \phi_\delta)\|_{L^\infty} \|\partial(\phi_\epsilon - \phi_\delta)\|_{L^0}^{1/2} \lesssim \epsilon^{-1} \epsilon^{1/2}$$
I am unable to generalize this to the case $u \in H^s$ or $s = 1/2$. 
Any suggestion or reference is greatly appreciated.


  [1]: https://math.stackexchange.com/questions/4512734/sobolev-space-is-spanned-by-distributions-supported-on-half-lines?noredirect=1#comment9476326_4512734
  [2]: https://math.mit.edu/%7Erbm/daomwc3.ps