When does this integral exist?

Question

I try to learn about Sobolev-spaces defined with tempered distribution.

I would like to understand for what $s\in\mathbb{R}$ does
$$\int (1+|\xi|^2)^s \mathrm d\xi\,, \qquad \xi \in \mathbb{R}^n$$ exist?

I think this could be helpful to understand Sobolev-spaces in this context. Can anybody help me with this? I am thankful for every help.

Answer 1

You function is continuous everywhere, hence you need to study what happens when $|\xi|\to\infty$. By comparison with $|\xi|^{2s}$ you deduce easily that for $2s+n < 0$ the integral converges and for $2s+n\ge0$ the integral diverges. Polar coordinates help, too.