I'm trying to solve a question from Folland's Real Analysis (9.26), and the only thing I am not able to prove is that $\widehat{G}(\xi ,\tau )=(2\pi i\tau +4\pi ^2 |\xi |^2)^{-1}$, $\widehat{G}: \mathbb{R}^n\times \mathbb{R}\to \mathbb{C}$, is $L^1_{loc}$.
The problem is around 0, as you can see.
The question asks to prove that this function is a tempered function, meaning it is $L^1_{loc}$ and a tempered distribution.
I could argue everything else, but not this bit. Any help is appreciated.
It is true at least for dimensions $n\geq 3$.
$$\int_{-\epsilon}^{\epsilon} \int_{B_{\epsilon}(0)} |\widehat{G(\xi ,\tau)}|d\xi d\tau$$
$$=\int_{-\epsilon}^{\epsilon} \int_{B_{\epsilon}(0)}\dfrac{1}{\sqrt{4\pi^2\tau^2+16\pi^4|\xi|^4}}$$ $$\leq \int_{-\epsilon}^{\epsilon} \int_{B_{\epsilon}(0)} \dfrac{1}{|\xi|^2}d\xi d\tau$$
Which converges if $n\geq 3$. The function is continuous elsewhere, so this is enough to show local integrability.