Hardy's Inequality for Laguerre Expansions of Hermite Type

TL;DR

This paper proves Hardy's inequality for Laguerre expansions of Hermite type in multiple dimensions, establishing sharp bounds and extending classical inequalities with new exponents and norms.

Contribution

It introduces the first multi-dimensional Hardy's inequality for Laguerre expansions of Hermite type with specific indices and sharpens the inequality using $L^1$ norms.

Findings

Hardy's inequality holds for the specified indices in multiple dimensions.

The sharp analogue with $L^1$ norm is established, with a slightly increased exponent.

The results extend classical inequalities to new settings with precise bounds.

Abstract

Hardy's inequality for Laguerre expansions of Hermite type with the index $\al\in(\{-1/2\}\cup[1/2,\infty))^d$ is proved in the multi-dimensional setting with the exponent $3d/4$. We also obtain the sharp analogue of Hardy's inequality with $L^1$ norm replacing $H^1$ norm at the expense of increasing the exponent by an arbitrarily small value.