Lusin characterisation of Hardy spaces associated with Hermite operators

Let $d\in \{3,4,5,\dots \}$ and $p\in (0,1]$. We consider the Hermite operator $L=-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$ 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) } $withrespecttothequasi-norm$ |\cdot|{H_L^p} = |$\mathcal{M}$\cdot|_{L^p}, $where$$\mathcal{M}$L f(\cdot) = \sup{t > 0} |e^{-tL} f(\cdot)|$forall$f \in L^2($\mathbb{R}$^d)$. We characterise $H_L^p({\mathbb{R}}^d)$ in terms of Lusin integrals associated with Hermite operator.