Laplacian

Consider the Laplacian on \(\mathbb{R}^{n}\) given by \[\begin{align} L = \frac{1}{4 \pi^{2}} \sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}} \end{align}\]

We then have the following identity for any admissible function \(f:\mathbb{R}^{n} \rightarrow \mathbb{R}\). Suppose \(f_1(x) = - \|x\|^{2}f(x)\), then

\[\begin{align} \mathcal{F}(f_1)(x) = L\hat{f}(t), \end{align}\] where \(\mathcal{F}\) is the Fourier transform.