Regularity of fractional Schr\"odinger 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 Schr\"odinger equation $i\partial_tu + (-\Delta)^\nu u = 0, u|_{t=0} = u_0$, for any $\nu > 0$, satisfies the Hardy space estimate that $$ \|u(t,\cdot)\|_{H^p(\mathbb H^d)} \leq C_p (1 + t)^{Q|1/p-1/2|}\|(1-\Delta)^{\nu Q|1/p-1/2|}u_0\|_{H^p(\mathbb H^d)}, $$ 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.