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$\delta$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_\nu f\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_\nu} \leq Ch^\beta\|f\|_{L^2_\nu}. \] 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 $\alpha=(\alpha_1,...,\alpha_d)\in[-1/2,\infty)^d$, we consider the Hermite-Laguerre operator $\mathcal{L}^\alpha=-\Delta+\arrowvert x\arrowvert^2+\sum_{i=1}^{d}(\alpha_j^2-\frac{1}{4})\frac{1}{x_i^2}$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $\mathcal{L}^\alpha$ which is defined as $S_R^{\lambda}(\mathcal{L}^\alpha)f(x)=\sum_{n=0}^{\infty}(1-\frac{4n+2\arrowvert\alpha\arrowvert_1+2d}{R^2})_{+}^{\lambda}\mathcal{P}_nf(x)$. Here $\mathcal{P}_nf(x)$ is the n-th Laguerre spectral projection operator and $\arrowvert\alpha\arrowvert_1$ denotes $\sum_{i=1}^{d}\alpha_i$. For $2\leq p<\infty$, we prove that \[ \lim_{R \to \infty} S_R^{\lambda}(\mathcal{L}^\alpha)f = f \quad \text{a.e.} \] 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.

math.FA↗

arXiv2025

Almost everywhere convergence of the convolution type Laguerre expansions

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 $\lambda>0$.

math.FA↗

arXiv2023

Observability and unique continuation inequalities for the Schr\"{o}dinger equations with inverse-square potentials

This paper is inspired by Wang, Wang and Zhang's work [ Observability and unique continuation inequalities for the Schr\"odinger equation. J. Eur. Math. Soc. 21, 3513--3572 (2019)], where they present several observability and unique continuation inequalities for the free Schr\"{o}dinger equation in $\mathbb{R}^{n}$. We extend all such observability and unique continuation inequalities for the Schr\"{o}dinger 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↗

arXiv2023

Uncertainty Principle and Geometric Condition for the Observability of Schr\"{o}dinger Equations

We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger 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 Schr\"odinger 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 $\Omega$, 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 $\Omega$, with a constant independent of the position of the interval.

math.AP↗