Lusin characterisation of Hardy spaces associated with Hermite operators

Abstract

Let $d\in \{3,4,5,\dots \}$ 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 $\Vert \cdot {\Vert }_{H_L^p}=\Vert \mathcal{M}\cdot {\Vert }_{L^p},$ 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.

16 pages