Three dimensional Fourier transform depending on a parameter

Question

Given a real parameter $\varepsilon>0$, consider the function $g_{\varepsilon}\in L^1(\mathbb{R}^3)$ given by $$g_{\varepsilon}(x)=\frac{1}{(|x|^2+\sqrt{\varepsilon})^2+\varepsilon}$$ The Fourier transform of $g_{\varepsilon}$ belongs to $L^{\infty}(\mathbb{R}^3)$ for any $\varepsilon>0$.

I want to show that $\Vert \hat{g_{\varepsilon}}\Vert_{L^{\infty}}$ diverges as $\varepsilon$ goes to zero.

Thank you for any suggestions.

Answer 1

You can just look this at $0$: we compute $$\| \hat{g_{\varepsilon}}\|_{L^{\infty}}\geq \hat{g_{\varepsilon}}(0)=\int_{\mathbb R}g_{\varepsilon}(x)\,dx=\int_{\mathbb R}\frac{1}{(|x|^2+\sqrt{\varepsilon})^2+\varepsilon}\,dx\xrightarrow[\varepsilon\to 0]{}\int_{\mathbb R}|x|^{-4}\,dx=\infty,$$ from the monotone convergence theorem.