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 $d{\mu }_{\alpha }(x)=x_1^{2{\alpha }_1+1}\cdot \cdot \cdot x_d^{{\alpha }_d}d{x}_1\cdot \cdot \cdot d{x}_d$. We consider the Laguerre operator $L_α=-Δ+\sum_{i=1}^{d}\frac{2α_j+1}{x_j}\frac{d}{dx_j}+\arrowvert x\arrowvert^2$ 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 $S_R^{\lambda }(L_{\alpha })f(x)={\sum }_{n=0}^{\infty }(1-\frac{e_n}{R^2}{)}_{+}^{\lambda }{P}_{n}f(x)$. 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 $2\le p<\infty$, we prove that $$\underset{R\to \infty }{\lim }{S}_R^{\lambda }(L_{\alpha })f=f-a.e.$$ for all $f\in L^p({\mathbb{R}}_{+}^d,d{\mu }_{\alpha }(x))$, provided that $\lambda >\lambda (\alpha ,p)/2$, where $λ(α,p)=\max\{2(\arrowvertα\arrowvert_1+d)(1/2-1/p)-1/2,0\}$, and $\arrowvertα\arrowvert_1:=\sum_{j=1}^{d}α_{j}$. Conversely, if $2\arrowvertα\arrowvert_{1}+2d>1$, 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 $(4\arrowvertα\arrowvert_{1}+4d)/(2\arrowvertα\arrowvert_{1}+2d-1)< p$ such that the convergence fails. When $2\arrowvertα\arrowvert_{1}+2d\leq1$, our results show that a.e. convergence holds for $f\in L^p({\mathbb{R}}_{+}^d,d{\mu }_{\alpha }(x))$ with $p\ge 2$ whenever $\lambda >0$.

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