Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that :
$$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$
Where $\gamma_s(|x-y|) = 2 (2 \pi)^{\frac{-n}{2}}[\displaystyle\int_{R^n} (1 + |\theta|^2) \sin^2 \displaystyle\frac{(x-y) . \theta}{2} \ d \theta] .$
I dont have idea how to prove that . Someone can give me a hint ?
I know a know something near in relation with my question and maybe helps:
for $f \in H^s(R^n)$ $(s> n/2)$ exists $\alpha \in (0,1)$ such that
$$| f(x+h) - f(x)| \leq 2^{1 - \alpha}|h|^{\alpha} (2 \pi)^{\frac{-n}{2}}[\displaystyle\int_{R^n} (1 + |\theta|^2)^{-(s - \alpha)} \ d \theta]$$
Thanks in advance