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

$L^p -L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces