  In this paper, we study H\"ormander type Fourier multiplier theorem and the
Nikolskii inequality on quantum tori. On the way to obtain these results, we
also prove some classical inequalities such as Paley type,
Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities
on quantum tori. As applications we establish embedding theorems between
Sobolev, Besov spaces as well as embeddings between Besov and Wiener and
Beurling spaces on quantum tori. We also analyse $\beta$-versions of Wiener and
Beurling spaces and their embeddings, and interpolation properties of all these
spaces on quantum tori. As an applications of the analysis, we also derive a
version of the Nash inequality, and the time decay for solutions of a heat type
equation.
