Let $k=2$ throughout.
Let $u, \Delta u \in L^2(\mathbb{R}^n)$ then $u \in H^k(\mathbb{R}^n)$ where $H^k= \{u \in S'(\mathbb{R}^n): (1+|\xi|^2)^{\frac{k}{2}} \widehat u \in L^2\}$.
I am a bit unsure about my answer so I'd appreciate any feedback:
By Plancherel's theorem and the Fourier differentiation rules $ \Delta u \in L^2 \implies \widehat{ \Delta u} = -|\xi|^2 \widehat u \in L^2$. But $\widehat u \in L^2$, again by Plancherel, so $(1+|\xi|^2) \widehat u \in L^2$, which concludes the proof.