Hardy's type inequality for the over critical exponent associated with the Dunkl transform

TL;DR

This paper improves Hardy's type inequality related to the Dunkl transform for functions in Hardy spaces, extending the inequality to a specific range of parameters with respect to measures homogeneous of degree .

Contribution

It extends Hardy's inequality for the Dunkl transform to a broader parameter range on Hardy spaces, providing sharper bounds and conditions.

Findings

Improved Hardy's inequality for Dunkl transform on Hardy spaces.

Extended the inequality to a wider parameter strip.

Established sharper bounds for the inequality.

Abstract

\begin{abstract} For the Hardy space $H^p(\mathbb{R}^{d})$, $ 0<p\leq 1,$ we shall improve a Hardy's type inequality associated with Dunkl transform respect to the measures $d\mu_{k}$ homogeneous of degree $\gamma ,$ on the strip $(2\gamma+d)(2-p)\leq\sigma<2\gamma+d+p(N+1),$ where $N =[(2\gamma+d)(1/p-1)]$ is the greatest integer not exceeding $(2\gamma+d)(1/ p -1).$ \end{abstract}