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Let $H_n$ be the Heisenberg group of dimension $2n+1$. Let ${\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n$ be the partial sub-Laplacians on $H_n$ and $T$ the central element of the Lie algebra of $H_n$. We prove that the kernel of the operator $m\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n,-iT\right)$ is in the Schwartz space $S\left(H_n\right)$ if $m\in S\left({\mathbb{R}}^{n+1}\right)$. We prove also that the kernel of the operator $h\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n\right)$ is in $S\left(H_n\right)$ if $h\in S\left({\mathbb{R}}^n\right)$ and that the kernel of the operator $g\left(\mathcal{L},-iT\right)$ is in $S\left(H_n\right)$ if $g\in S\left({\mathbb{R}}^2\right)$. Here $\mathcal{L}={\mathcal{L}}_1+\mathrm{\dots }+{\mathcal{L}}_n$ is the Kohn-Laplacian on $H_n$.

We prove Strichartz inequalities for the solution of the Schrödinger equation related to the full Laplacian on the Heisenberg group. A key point consists in estimating the decay in time of the $L^{\infty }$ norm of the free solution; this requires a careful analysis due also to the non-homogeneous nature of the full Laplacian.