Dirac distribution and Sobolev spaces

Question

I see the conclusion that

$\delta_{x_{0}} \in H^{s}\left(\mathbb{R}^{n}\right)$ if and only if $s<-n/2$, where $ H^{s}\left(\mathbb{R}^{n}\right) $ is Sobolev space.

from many places, and the hint is polar coordinates or Sobolev embedding. How can I use the hint to prove the conclusion? Thanks very much.

Answer 1

The easiest way is to use Fourier transform. The Fourier transform of $\delta_{x_0}$ is $y\mapsto e^{-ix_0\cdot y}$. Therefore $$ \|\delta_0\|_{H^s}^2 = \int_{\mathbb{R}^n} |e^{-ix_0\cdot y}|^2\,(1+|y|^2)^{s}\,\mathrm{d}y = \int_{\mathbb{R}^n} (1+|y|^2)^{s}\,\mathrm{d}y $$ which is finite if and only if $s<-n/2$.