$\newcommand\ep\epsilon\newcommand\Om\Omega\newcommand\al\alpha\newcommand\R{\mathbb R}$According to your comment, $C^{0,2s+\ep}(\Om)$ is the set of all functions on $\Om$ that are Hölder-continuous with exponent $2s+\ep\in(0,1)$. It also appears that you extend the functions $u\in C^{0,2s+\ep}(\Omega)$ to $\R^n$ by letting $u:=0$ on $\R^n\setminus\Om$.
If such an extended function $u$ is Hölder-continuous on $\R^n$ with exponent $2s+\ep$, then there is some real $c>0$ such that $$|u(x)-u(y)|\le c|x-y|^{2s+\ep}$$ for all $x,y$ in $\R^n$, whence for $D$ defined as the diameter of $\Om$ and all $x\in\Om$ $$\int_{\R^n}\frac{|u(x)-u(y)|}{|x-y|^{n+2s}}\,dy \ll\int_0^D \frac{r^{2s+\ep}}{r^{n+2s}}r^{n-1}\,dr=\int_0^D r^{\ep-1}\,dr<\infty$$ provided that $\ep>0$, as desired.