Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian

Abstract

Let $\mathcal{L}$ denote the twisted Laplacian in ${\mathbb{C}}^d$. We study almost everywhere convergence of the Bochner--Riesz mean $S_t^{\delta }(\mathcal{L})f$ of $f\in L^p({\mathbb{C}}^d)$ as $t\to \infty$, which is an expansion of $f$ in the special Hermite functions. For $2\le p\le \infty$, we obtain the sharp range of the summability indices $\delta$ for which the convergence of $S_t^{\delta }(\mathcal{L})f$ holds for all $f\in L^p({\mathbb{C}}^d)$.

23 pages