A doubt. Let $u$ in $L^p(\mathbb{R}^n)$ (then $u\in S'(\mathbb{R^n}$) the set of tempered functions) and $\mathcal{F}$ is the Fourier transform.
I know that $\mathcal{F}u\in S'(\mathbb{R}^n)$ but $(1+|x|^2e^{|x|^2})\mathcal{F}u\in S'(\mathbb{R}^n)$? (or $(1+|x|^2e^{|x|^2})\mathcal{F}u\in L^p(\mathbb{R}^n)?$)