Hardy's inequality for Hermite expansions revisited

TL;DR

This paper provides a simplified proof of Hardy's inequality for Hermite expansions in Hardy spaces, introducing a new estimate in one dimension for the range 0<p<1, using atomic decomposition techniques.

Contribution

It offers a concise proof of Hardy's inequality for Hermite expansions and extends the results to a new estimate in one dimension for 0<p<1.

Findings

Simplified proof of Hardy's inequality for Hermite expansions.

New estimate of Hardy's inequality in one dimension for 0<p<1.

Application of atomic decomposition in Hardy spaces associated with Hermite operators.

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 $0<p<1.$