On Schrödinger Groups of Fractional Powers of Hermite Operators

Abstract

Abstract Let $\mathcal {H}=-\Delta + |x|^2$ be the Hermite operator on $\mathbb R^n$ with $n\ge 2$. In this paper, we prove the boundedness of Schrödinger groups of fractional powers of $\mathcal {H}$ on Lebesgue and Hardy spaces. More precisely, we prove that (a) for $p \in (1,\infty )$, $\gamma>0$ and $\beta /\gamma \geq n|1/p -1/2|$, $$ \begin{align*} \big\|\mathcal{H}^{-\beta /2}e^{it \mathcal{H}^{\gamma /2}}f\big\|_{L^p(\mathbb R^n)} \leq C (1 + |t|)^{n|1/p-1/2|}\|f\|_{L^p(\mathbb R^n)}, \quad \forall t \in \mathbb{R}, \end{align*}$$and (b) for $p \in (0, 1]$, $\gamma>0$ and $\beta /\gamma \geq n(1/p -1/2)$, $$ \begin{align*} \big\|\mathcal{H}^{-\beta /2}e^{it \mathcal{H}^{\gamma /2}}f\big\|_{H^p_{\mathcal{H}}(\mathbb R^n)} \leq C (1 + |t|)^{n(1/p-1/2)}\|f\|_{H^p_{\mathcal{H}}(\mathbb R^n)}, \quad \forall t \in \mathbb{R}, \end{align*}$$where $H^p_{\mathcal {H}}(\mathbb R^n)$ is the Hardy space associated with the operator $\mathcal {H}$. These estimates improve related result of Thangavelu [22] and have some interesting applications.