Regularity of fractional Schrödinger equations and sub-Laplacian multipliers on the Heisenberg group

We define functions of the sub-Laplacian $\mathrm{\Delta }$ on the Heisenberg group ${\mathbb{H}}^d$ as Fourier multipliers. In this setting, we show that the solution $u$ of the free fractional Schrödinger equation $i{\mathrm{\partial }}_{t}u+(-\mathrm{\Delta }{)}^{\nu }u=0,u{\mathrm{\mid }}_{t=0}=u_0$, for any $\nu >0$, satisfies the Hardy space estimate that $$\mathrm{\parallel }u(t,\cdot ){\mathrm{\parallel }}_{H^p({\mathbb{H}}^d)}\le C_p(1+t{)}^{Q\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }}\mathrm{\parallel }(1-\mathrm{\Delta }{)}^{\nu Q\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }}{u}_0{\mathrm{\parallel }}_{H^p({\mathbb{H}}^d)},$$ with $Q=2d+2$, for all $p\in (0,\mathrm{\infty })$, and the corresponding estimate with $p=\mathrm{\infty }$ in $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{H}}^d)$. This is done via a general regularity result for parameter dependent sub-Laplacian Fourier multipliers. We prove also that Bessel potential spaces on the Heisenberg group correspond to Sobolev spaces in the same way as in Euclidean space, also for Hardy spaces.