Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\alpha \in [-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validates that the known version of Hardy's inequality for the Hermite functions is sharp.
"On Hardy's Inequality for Hermite Expansions." Taiwanese J. Math. 24 (2) 301 - 315, April, 2020. https://doi.org/10.11650/tjm/190601