Hardy's inequality for Hermite expansions revisited

Abstract

In this article, we give a short proof of Hardy's inequality for Hermite expansions of functions in the classical Hardy spaces $H^p({\mathbb{R}}^n)$, by using an atomic decomposition of the Hardy spaces associated with the Hermite operators. When the space dimension is $1$, we obtain a new estimate of Hardy's inequality for Hermite expansions in $H^p(\mathbb{R})$ for the range $0<p<1.$