  In this paper, we prove several versions of the classical Paley inequality
for the Weyl transform. As an application, we discuss $L^p$-$L^q$ boundedness
of the Weyl multipliers and prove a version of the H\"ormander's multiplier
theorem. We also prove Hardy-Littlewood inequality. Finally, we study
vector-valued versions of these inequalities. In particular, we consider the
inequalities of Paley, Hausdorff-Young, and Hardy-Littlewood and their
relations.
