First of all, as explained in my comment, this is the same as asking if the Hilbert transform is bounded on $L^2(\mathbb R, w\, dx)$, with $w(x)=(1+|x|)^{2s}$. Or, to state this one more time, this reformulation follows because restricting $u$ to a half line is essentially the same as applying the Hilbert transform to $\widehat{u}$. Hunt, Muckenhoupt, Wheeden [proved][1] that this is equivalent to $w\in A_2$ (see Theorem 9 there).

It is easy to check, using the defining condition
$$
\int_I w \int_I w^{-1}\lesssim |I|^2
$$
of $A_2$, that indeed $w\in A_2$ for our weight $w(x)=(1+|x|)^{2s}$ as long as $-1/2<s<1/2$.

The argument cannot handle $s=1/2$ (or $s\le -1/2$), which one would perhaps also expect to be more difficult since it is the borderline case. For $s>1/2$, $H^s$ functions are continuous, so the result clearly no longer holds in this case.


  [1]: https://www.ams.org/journals/tran/1973-176-00/S0002-9947-1973-0312139-8/S0002-9947-1973-0312139-8.pdf