For a fixed d-tuple \(\alpha =(\alpha _1,\dots ,\alpha _d)\in (-1,\infty )^d\) , consider the product space \(\mathbb {R}_+^d{:}{=}(0,\infty )^d\) equipped with Euclidean distance \(\arrowvert \cdot \arrowvert \) and the measure \(d\mu _{\alpha }(x)=x_1^{2\alpha _1+1}\cdots x_{d}^{2\alpha _d+1}dx_1\cdots dx_d\) . We consider the Laguerre operator \(L_{\alpha }=-\Delta +\sum _{i=1}^{d}\frac{2\alpha _j+1}{x_j}\frac{d}{dx_j}+\arrowvert x\arrowvert ^2\) which is a positive, self-adjoint operator on \(L^2(\mathbb {R}_+^d,d\mu _{\alpha }(x))\) with compact resolvent. In this paper, we study almost everywhere convergence of the Bochner-Riesz means associated with \(L_\alpha \) which is defined by \(S_R^{\lambda }(L_\alpha )f(x)=\sum _{n=0}^{\infty }(1-\frac{e_n}{R^2})_{+}^{\lambda }P_nf(x)\) . Here \(e_n\) is n-th eigenvalue of \(L_{\alpha }\) , and \(P_nf(x)\) is the n-th Laguerre spectral projection operator. This corresponds to the convolution-type Laguerre expansions introduced in Thangavelu’s lecture (Princeton Univ. Press, Princeton, NJ, 1993). The fact that the kernel of this type of Laguerre expansion is a radial function suggests that we can examine the sharpness of summability indices by radial functions. For \(2\le p<\infty \) , we prove that \(\begin{aligned}\lim _{R\rightarrow \infty } S_R^{\lambda }(L_\alpha )f=f\quad \text {a.e.}\end{aligned}\) for all \(f\in L^p(\mathbb {R}_+^d,d\mu _{\alpha }(x))\) , provided that \(\lambda >\lambda (\alpha ,p)/2\) , where \(\lambda (\alpha ,p)=\max \{2(\arrowvert \alpha \arrowvert _1+d)(1/2-1/p)-1/2,0\}\) , and \(\arrowvert \alpha \arrowvert _1:=\sum _{j=1}^{d}\alpha _{j}\) . Conversely, if \(2\arrowvert \alpha \arrowvert _{1}+2d>1\) , we will show the convergence generally fails if \(\lambda <\lambda (\alpha ,p)/2\) in the sense that there is an \(f\in L^p(\mathbb {R}_+^d,d\mu _{\alpha }(x))\) for \((4\arrowvert \alpha \arrowvert _{1}+4d)/(2\arrowvert \alpha \arrowvert _{1}+2d-1)< p\) such that the convergence fails. When \(2\arrowvert \alpha \arrowvert _{1}+2d\le 1\) , 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 \(\lambda >0\) .
