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

\begin{abstract} For the Hardy space $H^p({\mathbb{R}}^d)$, $0<p\le 1,$ 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 $(2\gamma +d)(2-p)\le \sigma <2\gamma +d+p(N+1),$ where $N=[(2\gamma +d)(1\mathrm{/}p-1)]$ is the greatest integer not exceeding $(2\gamma +d)(1\mathrm{/}p-1)\mathrm{.}$ \end{abstract}