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

Abstract

Consider the space ${\mathbb{R}}_{+}^d=(0,\infty {)}^d$ equipped with Euclidean distance and the Lebesgue measure. For every $\alpha =({\alpha }_1,...,{\alpha }_d)\in [-1/2,\infty {)}^d$, we consider the Hermite-Laguerre operator $\mathcal{L}^α=-Δ+\arrowvert x\arrowvert^2+\sum_{i=1}^{d}(α_j^2-\frac{1}{4})\frac{1}{x_i^2}$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with ${\mathcal{L}}^{\alpha }$ which is defined as $S_R^λ(\mathcal{L}^α)f(x)=\sum_{n=0}^{\infty}(1-\frac{4n+2\arrowvertα\arrowvert_1+2d}{R^2})_{+}^λ\mathcal{P}_nf(x)$. Here ${\mathcal{P}}_{n}f(x)$ is the n-th Laguerre spectral projection operator and $\arrowvertα\arrowvert_1$ denotes ${\sum }_{i=1}^d{\alpha }_i$. For $2\le p<\infty$, we prove that \[ \lim_{R \to \infty} S_R^λ(\mathcal{L}^α)f = f \quad \text{a.e.} \] for all $f\in L^p({\mathbb{R}}_{+}^d)$ provided that $\lambda >\lambda (p)/2$ and $\lambda (p)=\max \{d(1/2-1/p)-1/2,0\}$. Conversely, we show that the convergence generally fails if $\lambda <\lambda (p)/2$ in the sense that there exists $f\in L^p({\mathbb{R}}_{+}^d)$ for $2d/(d-1)<p$ such that the convergence fails.

31 pages