Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let $H = -\Delta + |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^{\lambda}(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^{\lambda} P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$\lim\limits_{R\to \infty} S_R^{\lambda}(H) f=f \ \ \ \text{a.e.}$$ for all $f\in L^p(\mathbb R^n)$ provided that $\lambda> \lambda(p)/2$ and $\lambda(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $\lambda< \lambda(p)/2$ in the sense that there is an $f\in L^p(\mathbb R^n)$ for $2n/(n-1)\le p$ such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For $n\geq 2$ and $p\ge 2$ our result tells that the critical summability index for a.e. convergence for $S_R^{\lambda}(H)$ is as small as only the \emph{half} of the critical index for a.e. convergence of the classical Bochner-Riesz means. When $n = 1$, we show a.e. convergence holds for $f\in L^p({\mathbb R})$ with $p\geq 2$ whenever $\lambda>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^{\lambda}(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.