Bochner-Riesz means for Laguerre expansions

Abstract

We investigate the $L^p$ norm convergence of Bochner-Riesz means $S_R^{\delta }f$, $f\in L^p({\mathbb{R}}^{+},d{\mu }_{\beta })$ associated to Laguerre expansions of type $\beta \ge -1/2$. For any given $1<p<\infty$ and $\beta >-1/2$, we provide a complete characterisation of the order $\delta$ for which these means are uniformly bounded on $L^p$ spaces. A particularly noteworthy phenomenon arises for $-1/3<\beta <0$, where the admissible $(p,\delta )$ region differs substantially from those for $-1/2\le \beta \le -1/3$ or $\beta \ge 0$. Interestingly, the convergence for $-1/3<\beta <0$ is an interaction between the ranges $-1/2\le \beta \le -1/3$ and $\beta \ge 0$.

arXiv comment42 pages