  Given primes $\ell\ne p$, we record here a $p$-adic valued Fourier theory on
a local field over $\mathbf{Q}_\ell$, which is developed under the perspective
of group schemes. As an application, by substituting rigid analysis for complex
analysis, it leads naturally to the $p$-adic local functional equation at
$\ell$, which strongly resembles the complex one in Tate's thesis.
