Almost everywhere convergence of the convolution type Laguerre expansions

For a fixed d-tuple $\alpha =({\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_d)\in (-1,\mathrm{\infty }{)}^d$, consider the product space ${\mathbb{R}}_{+}^d:=(0,\mathrm{\infty }{)}^d$ equipped with Euclidean distance $\text{arrowvert}\cdot \text{arrowvert}$ and the measure $d{\mu }_{\alpha }(x)=x_1^{2{\alpha }_1+1}\cdot \cdot \cdot x_d^{{\alpha }_d}d{x}_1\cdot \cdot \cdot d{x}_d$. We consider the Laguerre operator $L_{\alpha }=-\mathrm{\Delta }+{\sum }_{i=1}^d\frac{2{\alpha }_j+1}{x_j}\frac{d}{d{x}_j}+\text{arrowvert}x{\text{arrowvert}}^2$ which is a compact, positive, self-adjoint operator on $L^2({\mathbb{R}}_{+}^d,d{\mu }_{\alpha }(x))$. In this paper, we study almost everywhere convergence of the Bochner-Riesz means associated with $L_{\alpha }$ which is defined by $S_R^{\lambda }(L_{\alpha })f(x)={\sum }_{n=0}^{\mathrm{\infty }}(1-\frac{e_n}{R^2}{)}_{+}^{\lambda }{P}_{n}f(x)$. Here $e_n$ is n-th eigenvalue of $L_{\alpha }$, and $P_{n}f(x)$ is the n-th Laguerre spectral projection operator. This corresponds to the convolution-type Laguerre expansions introduced in Thangavelu’s lecture \cite{TS3}. For $2\le p<\mathrm{\infty }$, we prove that $$\underset{R\to \mathrm{\infty }}{\lim }{S}_R^{\lambda }(L_{\alpha })f=f-a\mathrm{.}e\mathrm{.}$$ for all $f\in L^p({\mathbb{R}}_{+}^d,d{\mu }_{\alpha }(x))$, provided that $\lambda >\lambda (\alpha ,p)\mathrm{/}2$, where $\lambda (\alpha ,p)=\max \{2(\text{arrowvert}\alpha {\text{arrowvert}}_1+d)(1\mathrm{/}2-1\mathrm{/}p)-1\mathrm{/}2,0\}$, and $\text{arrowvert}\alpha {\text{arrowvert}}_1:={\sum }_{j=1}^d{\alpha }_j$. Conversely, if $2\text{arrowvert}\alpha {\text{arrowvert}}_1+2d>1$, we will show the convergence generally fails if $\lambda <\lambda (\alpha ,p)\mathrm{/}2$ in the sense that there is an $f\in L^p({\mathbb{R}}_{+}^d,d{\mu }_{\alpha }(x))$ for $(4\text{arrowvert}\alpha {\text{arrowvert}}_1+4d)\mathrm{/}(2\text{arrowvert}\alpha {\text{arrowvert}}_1+2d-1)<p$ such that the convergence fails. When $2\text{arrowvert}\alpha {\text{arrowvert}}_1+2d\le 1$, our results show that a.e. convergence holds for $f\in L^p({\mathbb{R}}_{+}^d,d{\mu }_{\alpha }(x))$ with $p\ge 2$ whenever $\lambda >0$.