Let $\delta_x$ denote the dirac distribution at $x \in \mathbb{R^n}$. I need to show that for $s=n/2+\alpha$ with $0\leq\alpha\leq1$,
$||\delta_x-\delta_y||\leq C_{\alpha}|x-y|^{\alpha}$ , $x,y \in \mathbb{R^n}$
Is this realted to some compactness result?
Without loss of generality, we may assume that $y=0$. $$ \widehat{\delta_x-\delta_0}(\xi)=e^{ix\cdot\xi}-1. $$ Then $$\begin{align} \|\delta_x-\delta_0\|_{H^{-s}}^2&=\int_{\mathbb{R}^n}(1+|\xi|^2)^{-s}|e^{ix\cdot\xi}-1|^2\,d\xi\\ &=4\int_{\mathbb{R}^n}(1+|\xi|^2)^{-s}\sin^2\frac{x\cdot\xi}{2}\,d\xi\\ &=4\int_{\mathbb{R}^n}\Bigl(1+\frac{|\eta|^2}{|x|^2}\Bigr)^{-s}\sin^2\frac{x\cdot\eta}{2\,|x|}\,\frac{d\eta}{|x|^n}\\ &=4\,|x|^{2\alpha}\int_{\mathbb{R}^n}\bigl(|x|^2+|\xi|^2\bigr)^{-s}\sin^2\frac{x\cdot\eta}{2\,|x|}\,d\eta. \end{align}$$ I leave to you to show that the last integral is bounded independently of $x$.
