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

Abstract

We define functions of the sub-Laplacian $Δ$ 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 $i\partial_tu + (-Δ)^νu = 0, u|_{t=0} = u_0$, for any $ν > 0$, satisfies the Hardy space estimate that $\|u(t,\cdot)\|_{H^p(\mathbb H^d)}$ is estimated from above by $(1 + t)^{Q|1/p-1/2|}\|(1-Δ)^{νQ|1/p-1/2|}u_0\|_{H^p(\mathbb H^d)}$, 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.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5