  We establish a number of uncertainty inequalities for the additive group of a
finite affine plane, showing that for $p$ prime, a nonzero function
$f\colon\mathbb F_p^2\to\mathbb C$ and its Fourier transform $\hat
f\colon\widehat{\mathbb F_p^2}\to\mathbb C$ cannot have small supports
simultaneously. The "baseline" of our investigation is the well-known
Meshulam's bound, which we sharpen, for the particular groups under
consideration, taking into account not only the sizes of the support sets
$\mathrm{supp}\,f$ and $\mathrm{supp}\,\hat f$, but also their structure.
  Our results imply in particular that, with some explicitly classified
exceptions, one has $|\mathrm{supp}\,f||\mathrm{supp}\,\hat f|\ge3p(p-2)$; in
comparison, the classical uncertainty inequality gives
$|\mathrm{supp}\,f||\mathrm{supp}\,\hat f|\ge p^2$.
