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

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-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 the logarithmic Sobolev and Nash type inequalities.