Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a constant $C>0$ such that: $$\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|}{|x-y|^{n+2s}}\,dy\leq C,\qquad\forall x\in\Omega,$$ with $C$ that not depend by $x\in\Omega$. Here $\epsilon>0$ is such that $2s+\epsilon\in(0,1)$, and for every $\alpha>0$, $C^{0,\alpha}(A)$ is the space of Holder continuous functions on $A\subset\mathbb{R}^n$. I have no idea on how to proceed, any help would be appreciated.

Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a constant $C>0$ such that: $$\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|}{|x-y|^{n+2s}}\,dy\leq C,\qquad\forall x\in\Omega,$$ with $C$ that not depend by $x\in\Omega$. Here $\epsilon>0$ is such that $2s+\epsilon\in(0,1)$, and for every $\alpha>0$, $C^{0,\alpha}(A)$ is the space of Holder continuous functions on $A\subset\mathbb{R}^n$. Under what assumptions about u is my claim true? I have no idea on how to proceed, any help would be appreciated.