Decay estimates for a class of wave equations on the Heisenberg group

In this paper, we study a class of dispersive wave equations on the Heisenberg group \(H^n\). Based on the group Fourier transform on \(H^n\), the properties of the Laguerre functions and the stationary phase lemma, we establish the decay estimates for a class of dispersive semigroup on \(H^n\) given by \(e^{\textrm{it}\phi ({\mathscr {L}})}\), where \(\phi : {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) is smooth, and \({\mathscr {L}}\) is the sub-Laplacian on \(H^n\). Finally, using the duality arguments, we apply the obtained results to derive the Strichartz inequalities for the solutions of some specific equations, such as the fractional Schrödinger equation, the fractional wave equation and the fourth-order Schrödinger equation.