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