Decay estimates for the linear damped wave equation on the Heisenberg group

This paper is devoted to the derivation of L2 - L2 decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group Hn, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further L1(Hn) regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of Hn and on the properties of the Hermite functions that form a maximal orthonormal system for L2(Rn) of eigenfunctions of the harmonic oscillator.