  In this paper, we study Fourier multipliers on quantum Euclidean spaces and
obtain results on their $L^p -L^q$ boundedness. On the way to get these
results, we prove Paley, Hausdorff-Young, Hausdorff-Young-Paley, and
Hardy-Littlewood inequalities on the quantum Euclidean space. As applications,
we establish the $L^p -L^q$ estimate for the heat semigroup and Sobolev
embedding theorem on quantum Euclidean spaces. We also obtain quantum analogues
of logarithmic Sobolev and Nash type inequalities.
