arXiv subjects

Wentao Teng

Publications and source records attributed to Wentao Teng.

4 recordsLinked to original sources

arXiv2026

A positive product formula of integral kernels of $k$-Hankel transforms

The $k$-Hankel transform $F_{k,1}$ (or the $(k,1)$-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in $(k,a)$-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure $σ_{x,t}^{k,1}(ξ)$. We will then study the representing measure $σ_{x,t}^{k,1}(ξ)$ and analyze the support of this measure, and derive a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis.

math.CA↗

arXiv2022

Hardy inequalities for fractional $(k,a)$-generalized harmonic oscillator

In this paper, we will define $a$-deformed Laguerre operators $L_{a,α}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),dμ_{a,α}\right)$. Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value $z=\frac{πi}2$, of the $(k,a)$-generalized Laguerre semigroup introduced by S. Ben Saïd, T. Kobayashi and B. Ørsted. And then we prove a Hardy inequality for fractional powers of the $a$-deformed Dunkl harmonic oscillator $\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a$ using this expansion. When $a=2$, the fractional Hardy inequality reduces to that of Dunkl--Hermite operators given by Ó. Ciaurri, L. Roncal and S. Thangavelu. The operators $L_{a,α}$ also give a tangible characterization of the radial part of the $(k,a)$-generalized Laguerre semigroup on each $k$-spherical component $\mathcal H_k^m\left(\mathbb{R}^N\right)$ for $λ_{k,a,m}:=\frac{2m+2\left\langle k\right\rangle+N-2}a\geq -1/2$ defined via decomposition of unitary representation.

math.CA↗

arXiv2022

Imaginary powers of $(k,1)$-generalized harmonic oscillator

In this paper we will define and investigate the imaginary powers $\left(-\triangle_{k,1}\right)^{-iσ},σ\in\mathbb{R}$ of the $(k,1)$-generalized harmonic oscillator $-\triangle_{k,1}=-\left\|x\right\|\triangle_k+\left\|x\right\|$ and prove the $L^p$-boundedness $(1<p<\infty)$ and weak $L^1$-boundedness of such operators. It is a parallel result to the $L^p$-boundedness $(1<p<\infty)$ and weak $L^1$-boundedness of the imaginary powers of the Dunkl harmonic oscillator $-\triangle_k+\left\|x\right\|^2$. To prove this result, we develop the Calderón--Zygmund theory adapted to the $(k,1)$-generalized setting by constructing the metric space of homogeneous type corresponding to the $(k,1)$-generalized setting, and show that $\left(-\triangle_{k,1}\right)^{-iσ}$ are singular integral operators satisfying the corresponding Hörmander type condition.

math.CA↗

arXiv2021

Dunkl translations, Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$

In this paper, we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$, and prove the boundedness of Riesz transforms from $L^\infty$ to Dunkl-type $BMO$ space under the uniform boundedness assumption of Dunkl translations. The proof and the definition in Dunkl setting will be harder than in the classical case for the lack of some similar properties of Dunkl translations to that of classical translations. We will also extend the preciseness of the description of support of Dunkl translations on characteristic functions by Gallardo and Rejeb to that on all nonnegative radial functions in $L^2(m_k)$.

math.FA↗