  In this paper we study the $L^{p}$-$L^{q}$ boundedness of Fourier multipliers
on the fundamental domain of a lattice in $\mathbb{R}^{d}$ for $1 < p,q <
\infty$ under the classical H\"ormander condition. First, we introduce Fourier
analysis on lattices and have a look at possible generalisations. We then prove
the Hausdorff-Young inequality, Paley's inequality and the
Hausdorff-Young-Paley inequality in the context of lattices. This amounts to a
quantitative version of the $L^{p}$-$L^{q}$ boundedness of Fourier multipliers.
Moreover, the Paley inequality allows us to prove the Hardy-Littlewood
inequality.
