Fourier transform over a local field ($\hat{f}(y)={\int }_{F}f(x)\psi (xy)dx$)

For a local field $F$, an additive character $\psi$, and a Haar measure $dx$, the Fourier transform isIt maps the Schwartz-Bruhat space to itself.

For an additive character $\psi$ of a non-Archimedean local field $F$,For the character induced from the standard character of ${\mathbb{Q}}_p$, this annihilator is the inverse different.