# real analysis – Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder

Let $$Omegasubsetmathbb{R}^n$$, let $$sin (1/2,1)$$, let $$uin C^{1,2s-1+epsilon}(Omega)$$ such that: $$u=0$$ on $$mathbb{R}^nsetminusOmega$$, and: $$uin C^{0,s}(mathbb{R}^n)$$, is true that there exist a constant $$C>0$$, such that:
$$int_{mathbb{R}^n}frac{|u(x)-u(y)|}{|x-y|^{n+2s}},dyleq C,qquadforall xinmathbb{R}^n.$$
Her $$C>0$$ does not depend to x.
Here $$epsilon>0$$ is small such that: $$2s-1+epsilonin(0,1)$$, and $$C^{k,alpha}(A)$$ is the space of $$alpha$$-Hölder functions whose derivative of order less than $$kinmathbb{N}$$ are $$alpha$$-Hölder on the open set $$Asubsetmathbb{R}^n$$. I have no idea on how to proceed, any help is appreciated.