In this paper, we study the structural damped wave equation on the Heisenberg group \({\mathbb {H}}_n\) and its solution \(u(t, \eta )\) , \(t> 0, \eta \in {\mathbb {H}}_n\) . We prove decay estimates including \(L^1-L^2\) estimates and \(L^1-L^\infty \) estimates for the solution \(u(t, \eta )\) and its higher-order time derivative and horizontal gradient \(\partial _t^\ell \nabla _{{\mathbb {H}}_n}^k u(t, \eta )\) . Our approach relies on some detailed estimates using the group Fourier transform and the functional calculus of the sub-Laplacian. The obtained results are not only new but also improve previously known results.
