Almost everywhere convergence of the convolution type Laguerre expansions
Longben Wei
TL;DR
The paper resolves almost everywhere convergence for Bochner-Riesz means S_R^{\lambda}(L_{\alpha}) tied to the Laguerre operator on \mathbb{R}_+^d. By developing weighted L^2 maximal and square-function estimates, along with sharp trace lemmas and Littlewood-Paley theory in the Laguerre setting, it establishes that a.e. convergence holds for 2 \le p < \infty when \lambda>\lambda(\alpha,p)/2, with precise sharpness statements and endpoint remarks. The methods blend spectral theory, weighted harmonic analysis, and finite-speed propagation, extending previous results for Hermite/twisted Laplacian contexts to the convolution-type Laguerre framework. These results are significant for understanding summability and pointwise convergence of Laguerre-type expansions in higher dimensions, providing concrete thresholds that govern a.e. convergence and its limitations. Overall, the work clarifies how spectral discreteness and weighted techniques yield improved summability indices in a Laguerre setting with potential applications to related orthogonal expansions.
Abstract
For a fixed d-tuple $α=(α_1,...,α_d)\in(-1,\infty)^d$, consider the product space $\mathbb{R}_+^d:=(0,\infty)^d$ equipped with Euclidean distance $\arrowvert \cdot \arrowvert$ and the measure $dμ_α(x)=x_1^{2α_1+1}\cdot\cdot\cdot x_{d}^{α_d}dx_1\cdot\cdot\cdot dx_d$. 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 $L^2(\mathbb{R}_+^d,dμ_α(x))$. In this paper, we study almost everywhere convergence of the Bochner-Riesz means associated with $L_α$ which is defined by $S_R^λ(L_α)f(x)=\sum_{n=0}^{\infty}(1-\frac{e_n}{R^2})_{+}^λP_nf(x)$. Here $e_n$ is n-th eigenvalue of $L_α$, 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 $$\lim_{R\rightarrow\infty} S_R^λ(L_α)f=f\,\,\,\,-a.e.$$ for all $f\in L^p(\mathbb{R}_+^d,dμ_α(x))$, provided that $λ>λ(α,p)/2$, 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 $λ<λ(α,p)/2$ in the sense that there is an $f\in L^p(\mathbb{R}_+^d,dμ_α(x))$ 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 $f\in L^p(\mathbb{R}_+^d,dμ_α(x))$ with $p\geq 2$ whenever $λ>0$.