  Let $d \in \{3, 4, 5, \ldots\}$ and $p \in (0,1]$. We consider the Hermite
operator $L = -\Delta + |x|^2$ on its maximal domain in $L^2(\mathbb{R}^d)$.
Let $H_L^p(\mathbb{R}^d)$ be the completion of $ \{ f \in L^2(\mathbb{R}^d):
\mathcal{M}_L f \in L^p(\mathbb{R}^d) \} $ with respect to the quasi-norm $
\|\cdot\|_{H_L^p} = \|\mathcal{M}\cdot\|_{L^p}, $ where $\mathcal{M}_L f(\cdot)
= \sup_{t > 0} |e^{-tL} f(\cdot)|$ for all $f \in L^2(\mathbb{R}^d)$. We
characterise $H_L^p(\mathbb{R}^d)$ in terms of Lusin integrals associated with
Hermite operator.
