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

Let $H=-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$ 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$P_k f$isthe$k$-thHermitespectralprojectionoperator\mathrm{.}For$2\le p<\infty$,weprovethat$$\underset{R\to \mathrm{\infty }}{\lim }{S}_R^{\lambda }(H)f=f\text{ a.e.}$$forall$f\in L^p(\mathbb R^n)$providedthat$\lambda> \lambda(p)/2$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 $\lambda <\lambda (p)\mathrm{/}2$ in the sense that there is an $f\in L^p({\mathbb{R}}^n)$ for $2n\mathrm{/}(n-1)\le p$ such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For $n\ge 2$ 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 $p\ge 2$ 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.