arXiv subjects

Longben Wei

Publications and source records attributed to Longben Wei.

5 recordsLinked to original sources

arXiv2026

Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are$δ$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_νf\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_ν} \leq Ch^β\|f\|_{L^2_ν}. \] The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.

math.CA↗

arXiv2025

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

Consider the space $\mathbb{R}_+^d=(0,\infty)^d$ equipped with Euclidean distance and the Lebesgue measure. For every $α=(α_1,...,α_d)\in[-1/2,\infty)^d$, 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 $\mathcal{L}^α$ 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 $\mathcal{P}_nf(x)$ is the n-th Laguerre spectral projection operator and $\arrowvertα\arrowvert_1$ denotes $\sum_{i=1}^{d}α_i$. For $2\leq p<\infty$, 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 $λ>λ(p)/2$ and $λ(p)=\max\{d(1/2-1/p)-1/2,0\}$. Conversely, we show that the convergence generally fails if $λ<λ(p)/2$ in the sense that there exists $f\in L^p({\mathbb{R}_+^d})$ for $2d/(d-1)< p$ such that the convergence fails.

math.FA↗

arXiv2025

Almost everywhere convergence of the convolution type Laguerre expansions

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 $λ>0$.

math.FA↗

arXiv2025

Uncertainty Principle and Geometric Condition for the Observability of Schrödinger Equations

We provide necessary and sufficient geometric conditions for the exact observability of the Schrödinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schrödinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $Ω$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $Ω$, with a constant independent of the position of the interval.

math.AP↗

arXiv2023

Observability and unique continuation inequalities for the Schrödinger equations with inverse-square potentials

This paper is inspired by Wang, Wang and Zhang's work [ Observability and unique continuation inequalities for the Schrödinger equation. J. Eur. Math. Soc. 21, 3513--3572 (2019)], where they present several observability and unique continuation inequalities for the free Schrödinger equation in $\mathbb{R}^{n}$. We extend all such observability and unique continuation inequalities for the Schrödinger equations on half-line with inverse-square potentials. Technically, the proofs essentially rely on the representation of the solution, a Nazarov type uncertainty principle for the Hankel transform and an interpolation inequality for functions whose Hankel transform have compact support.

math.AP↗