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

Abstract

We define functions of the sub-Laplacian $\Delta$ 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 $i{\partial }_{t}u+(-\Delta {)}^{\nu }u=0,u{|}_{t=0}=u_0$, for any $\nu >0$, satisfies the Hardy space estimate that $$\Vert u(t,\cdot ){\Vert }_{H^p({\mathbb{H}}^d)}\le C_p(1+t{)}^{Q|1/p-1/2|}\Vert (1-\Delta {)}^{\nu Q|1/p-1/2|}{u}_0{\Vert }_{H^p({\mathbb{H}}^d)},$$ with $Q=2d+2$, for all $p\in (0,\infty )$, and the corresponding estimate with $p=\infty$ in $\mathrm{BMO}({\mathbb{H}}^d)$. 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.