Laplace operator and Fourier transform

Question

Let $f$ be a function with compact support in $D\subset \mathbf{R}^2$. Is is known that $$-\Delta f=F^{-1}|x|^2 F f,$$ where $F$ is the Fourier transform. Which is the $n-$dimensional analogous formula?

Answer 1

As long as $f$ does not have any terrible singularities, it is a tempered distribution. Therefore we have $$ (f, -\Delta \varphi) = (f, F^{-1}|\xi|^2F\varphi) $$ when $\varphi$ is a Schwartz function (a function of rapid decay whose derivatives also decay rapidly). Thus $$ -\Delta f = F^{-1}|\xi|^2F f $$ where the operators $\Delta$, $F$, and $F^{-1}$ are interpreted in the sense of distributions. This is perhaps a vacuous statement, but the point is that, so long as $\Delta f$ is defined, you can write it in terms of the Fourier transform.