Let us assume we have the following extension operator:
$$
\operatorname{ext}_R^\sigma u=
\begin{cases}
u(x) & \text{if }x \in (0,T)\\
u(0) & \text{if }x \in(0,T)^c.
\end{cases}
$$ We assume the fractional Sobolev space $W^{p,s}$, $s\in (0,1)$.
For the definition of the norm see p. 13 of the book by Giovanni Leoni, A first course in fractional Sobolev spaces, Graduate Studies in Mathematics 229, AMS, (2023), MR4567945, Zbl 1517.46001.
My question is how do we define a space $W$ such that $\operatorname{ext}_R^\sigma\in W^{p,s}(R)$?
I tried $u\in W^{p,s}(0,T)$ but this is impossible because the boundary term $\int_{(0,T)^c}u(0) \, dx$ is not finite and we also will get the other two terms when we calculate the fractional seminorm. How do we get rid of these boundary terms? I appreciate if you can provided any clues.
