The Inverse Fourier Transform $\mathcal{F}^{-1}(|\xi|^k)\in L^1$

Question

I was trying this question asked earlier today: Inverse of a Fourier Transform is $L^1 \cap L^{\infty}$

and I got the answer as long as I can show one thing:

$\mathcal{F}^{-1}(|\xi|^k)\in L^1(\mathbb{R^2})$, with $k>0.$

i.e. the Inverse Fourier Transform of $|\xi|^k\in L^1(\mathbb{R}^2)$.

(note that here $\xi=(\xi_1,\xi_2)$ and $|\xi|^k=(\xi_1^2+\xi_2^2)^{k/2})$

Anyone know how to do it?