Almost everywhere convergence of the convolution type Laguerre expansions

Abstract

For a fixed d-tuple $\alpha=(\alpha_1,...,\alpha_d)\in(-1,\infty)^d$, consider the product space $\mathbb{R}_+^d:=(0,\infty)^d$ equipped with Euclidean distance $\arrowvert \cdot \arrowvert$ and the measure $d\mu_{\alpha}(x)=x_1^{2\alpha_1+1}\cdot\cdot\cdot x_{d}^{\alpha_d}dx_1\cdot\cdot\cdot dx_d$. 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 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 $S_R^{\lambda}(L_\alpha)f(x)=\sum_{n=0}^{\infty}(1-\frac{e_n}{R^2})_{+}^{\lambda}P_nf(x)$. 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 \cite{TS3}. For $2\leq p<\infty$, we prove that $$\lim_{R\rightarrow\infty} S_R^{\lambda}(L_\alpha)f=f\,\,\,\,-a.e.$$ 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 $2\arrowvert\alpha\arrowvert_{1}+2d\leq1$, our results show that a.e. convergence holds for $f\in L^p(\mathbb{R}_+^d,d\mu_{\alpha}(x))$ with $p\geq 2$ whenever $\lambda>0$.