Hardy's inequalities for Hermite and Laguerre expansions revisited

Abstract

We show that Hardy's inequalities for Laguerre expansions hold on the space $L^1(0,\infty)$ when the Laguerre parameters $\alpha$ are positive, and we prove that although the inequality holds on the real Hardy space $H^1(0,\infty)$ if $\alpha= 0$, it does not hold on $L^1(0,\infty)$. Further, Hardy's inequality for Hermite expansion is established on $L^1(0,\infty)$.