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

$L^{p}$-$L^{q}$ boundedness of Fourier multipliers on Fundamental domains of Lattices in $\mathbb{R}^d$