I'm trying to show that the dirac delta function is in $H^{\frac{-n}{2}- \epsilon}(\mathbb{R}^{n}) \forall \epsilon > 0.$ Where $H^{s}(\mathbb{R}^{n})$ denotes Sobolev space of order $s$ on $\mathbb{R}^{n}$
I have that since $\hat\delta = 1 \Rightarrow <\xi>^{s}\hat\delta = <\xi>^{s}$ trivially.
So this amounts to needing to show that $\int_{\mathbb{R}^{n}} |(1+|\xi|^{2})^{\frac{s}{2}}|^{2}d \xi \in L^{2}(\mathbb{R}^{n}, s= -\frac{n}{2} - \epsilon.$
But now I'm stuck. I know I need to show that this integral is finite.
Any help appreciated.
Thanks.