# On Hardy's inequality for Hermite expansions

## TL;DR

This paper establishes a sharp multi-dimensional Hardy's inequality for Hermite-type Laguerre functions and generalized Hermite expansions, confirming the optimality of existing inequalities for Hermite functions.

## Contribution

It extends Hardy's inequality to multi-dimensional Hermite-type Laguerre functions and generalized Hermite expansions, demonstrating the sharpness of known inequalities.

## Key findings

- Proved sharp Hardy's inequality for Hermite-type Laguerre functions in multiple dimensions.
- Derived the corresponding inequality for generalized Hermite expansions.
- Confirmed the optimality of the existing Hardy's inequality for Hermite functions.

## Abstract

Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\al\in[-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validate that the known version of Hardy's inequality for the Hermite functions is sharp.