Regularity of fractional Schr\"odinger equations and sub-Laplacian multipliers on the Heisenberg group

TL;DR

This paper studies fractional Schr"odinger equations on the Heisenberg group, establishing Hardy space estimates and regularity results for sub-Laplacian Fourier multipliers, extending Euclidean space concepts to this non-commutative setting.

Contribution

It introduces a general regularity framework for parameter-dependent sub-Laplacian Fourier multipliers on the Heisenberg group, linking Bessel potential and Sobolev spaces.

Findings

Solutions satisfy Hardy space estimates with explicit time growth.

Bessel potential spaces on the Heisenberg group match Sobolev spaces as in Euclidean space.

Established regularity results for fractional Schr"odinger equations 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.