Almost everywhere convergence of Bochner-Riesz means for the Hermite type Laguerre expansions

Consider the space ${\mathbb{R}}_{+}^d=(0,\mathrm{\infty }{)}^d$ equipped with Euclidean distance and the Lebesgue measure. For every $\alpha =({\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_d)\in [-1\mathrm{/}2,\mathrm{\infty }{)}^d$, we consider the Hermite-Laguerre operator ${\mathcal{L}}^{\alpha }=-\mathrm{\Delta }+\text{arrowvert}x{\text{arrowvert}}^2+{\sum }_{i=1}^d({\alpha }_j^2-\frac{1}{4})\frac{1}{x_i^2}$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with ${\mathcal{L}}^{\alpha }$ which is defined as $S_R^{\lambda }({\mathcal{L}}^{\alpha })f(x)={\sum }_{n=0}^{\mathrm{\infty }}(1-\frac{4n+2\text{arrowvert}\alpha {\text{arrowvert}}_1+2d}{R^2}{)}_{+}^{\lambda }{\mathcal{P}}_{n}f(x)$. Here ${\mathcal{P}}_{n}f(x)$ is the n-th Laguerre spectral projection operator and $\text{arrowvert}\alpha {\text{arrowvert}}_1$ denotes ${\sum }_{i=1}^d{\alpha }_i$. For $2\le p<\mathrm{\infty }$, we prove that $$\underset{R\to \mathrm{\infty }}{\lim }{S}_R^{\lambda }({\mathcal{L}}^{\alpha })f=f\text{a.e.}$$ for all $f\in L^p({\mathbb{R}}_{+}^d)$ provided that $\lambda >\lambda (p)\mathrm{/}2$ and $\lambda (p)=\max \{d(1\mathrm{/}2-1\mathrm{/}p)-1\mathrm{/}2,0\}$. Conversely, we show that the convergence generally fails if $\lambda <\lambda (p)\mathrm{/}2$ in the sense that there exists $f\in L^p({\mathbb{R}}_{+}^d)$ for $2d\mathrm{/}(d-1)<p$ such that the convergence fails.