Almost Everywhere Convergence of Inverse Dunkl Transform on the Real Line

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