Almost Everywhere Convergence of Inverse Dunkl Transform on the Real Line

In this paper, we will first show that the maximal operator $S_{\ast }^{\alpha }$ of spherical partial sums $S_R^{\alpha }$, associated to Dunkl transform on $\mathbb{R}$ is bounded on $L^p(\mathbb{R},\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha +1}dx)$ functions when $\frac{4(\alpha +1)}{2\alpha +3}<p<\frac{4(\alpha +1)}{2\alpha +1}$, and it implies that, for every $L^p(\mathbb{R},\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha +1}dx)$ function $f(x)$, $S_R^{\alpha }f(x)$ converges to $f(x)$ almost everywhere as $R\to \mathrm{\infty }$. On the other hand we obtain a sharp version by showing that $S_{\ast }^{\alpha }$ is bounded from the Lorentz space $L^{p_i,1}(\mathbb{R},\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha +1})$ into $L^{p_i,\mathrm{\infty }}(\mathbb{R},\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha +1}),i=0,1$ where $p_0=\frac{4(\alpha +1)}{2\alpha +3}$ and $p_1=\frac{4(\alpha +1)}{2\alpha +1}$.