Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there exist a constant $C>0$, such that: $$ \int_{\mathbb{R}^n}\frac{|u(x)-u(y)|}{|x-y|^{n+2s}}\,dy\leq C,\qquad\forall x\in\mathbb{R}^n.$$ Her $C>0$ does not depend to x. Here $\epsilon>0$ is small such that: $2s-1+\epsilon\in(0,1)$, and $C^{k,\alpha}(A)$ is the space of $\alpha$-Hölder functions whose derivative of order less than $k\in\mathbb{N}$ are $\alpha$-Hölder on the open set $A\subset\mathbb{R}^n$. I have no idea on how to proceed, any help is appreciated.
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2$\begingroup$ Come on, $|u(x) - u(y)|$ is at least as large as $C |x - y|$ on a large set (unless $\nabla u(x) = 0$), and $C |x - y| / |x - y|^{n + 2s}$ is clearly non-integrable. $\endgroup$– Mateusz KwaśnickiNov 16, 2020 at 23:09
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$\begingroup$ @ Mateusz Kwaśnicki: can you help me with this question link, please? In this question there is what really i have to prove. Help me, please. $\endgroup$– inocNov 17, 2020 at 6:41
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