Riesz transforms for the Dunkl harmonic oscillator

Dunkl 设置下的分数阶 Riesz 变换及其交换子

The fractional Riesz transform and their commutator in Dunkl setting

本文研究 Dunkl 设置下分数阶 Riesz 变换的有界性，并给出其交换子在中心 BMO 与 Dunkl BMO 空间有界性的充要条件，进而用 VMO 空间刻画交换子的紧性。

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 descripti…

Riesz distributions and Laplace transform in the Dunkl setting of type A

We study Riesz distributions in the framework of rational Dunkl theory associated with root systems of type A. As an important tool, we employ a Laplace transform involving the associated Dunkl kernel, which essentially goes back to Macdonald, but was so far only established at a formal level. We give a rigorous treatment of this transform based on suitable estimates of the type A Dunkl kernel. Our main result is a precise analogue in the Dunkl setting of a well-known result by Gindikin, stating that a Riesz distribution on a symmetric cone is a positive measure if and only if its exponent is …

鞅变换与补偿Bellman估计在Dunkl Riesz变换中的应用

Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms

本文提出鞅变换框架，通过补偿Bellman函数证明Dunkl Riesz变换的$L^p$估计，并给出向量值估计及$G$-不变函数情形下的根系无关结果。

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those $1\le p,q \le \infty$, for which the potential operators satisfy $L^p-L^q$ estimates. In case of the Riesz potentials, we also characterize those $1\le p,q \le \infty$, for which two-weight $L^p-L^q$ estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight $L^p-L^q$ bou…

Imaginary Powers of the Dunkl Harmonic Oscillator

In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on $\mathbb{R}^d$ isomorphic to $\mathbb{Z}^d_2$. We prove that imaginary powers of this operator are bounded on $L^p$, $1<p<\infty$, and from $L^1$ into weak $L^1$.

Convolution operator and maximal function for Dunkl transform

For a family of weight functions, $h_κ$, invariant under a finite reflection group on $\RR^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.

Dunkl框架下的Hankel变换、K-Bessel函数与zeta分布

Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting

该研究在Dunkl框架下探究Dunkl型Hankel变换的解析性质，引入Bessel函数与K-Bessel函数的Dunkl类似物，研究zeta积分的分布推广及正则性，证明关联zeta分布与Dunkl变换的函数方程。

Inversion Formulas for the Spherical Radon-Dunkl Transform

The spherical Radon-Dunkl transform $R_κ$, associated to weight functions invariant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of $h$-harmonics. Several inversion formulas of $R_κ$ are given with the aid of spherical Riesz-Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.

Dunkl-Riesz变换向量值的无维弱型端点估计

Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform

本文提出Dunkl框架下的障碍/部分平衡方法，证明向量Dunkl-Riesz变换及其Schrödinger变体的无维弱型(1,1)估计，常数依赖反射群正根计数。

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. …

Multipliers of Laplace transform type in certain Dunkl and Laguerre settings

We investigate Laplace type and Laplace-Stieltjes type multipliers in the $d$-dimensional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphic to $\mathbb{Z}_2^d$ and in the related context of Laguerre function expansions of convolution type. We use Calderón-Zygmund theory to prove that these multiplier operators are bounded on weighted $L^p$, $1<p<\infty$, and from $L^1$ to weak $L^1$.

关于特定 Dunkl 和 Bessel 背景下的基本调和分析算子

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

本文在反射群同构于 $\mathbb{Z}_2^n$ 的 Dunkl Laplacian 多维背景下，研究了极大算子、$g$-函数、Riesz 变换等调和分析算子，并运用 Calderón-Zygmund 理论证明了其在加权 $L^p$ 空间及弱 $L^1$ 空间的有界性。

The Riesz transform for the harmonic oscillator in spherical coordinates

In this paper we show weighted estimates in mixed norm spaces for the Riesz transform associated with the harmonic oscillator in spherical coordinates. In order to prove the result we need a weighted inequality for a vector-valued extension of the Riesz transform related to the Laguerre expansions which is of independent interest. The main tools to obtain such extension are a weighted inequality for the Riesz transform independent of the order of the involved Laguerre functions and an appropriate adaptation of Rubio de Francia's extrapolation theorem.

Pitt's inequalities and uncertainty principle for generalized Fourier transform

We study the two-parameter family of unitary operators \[ \mathcal{F}_{k,a}=\exp\Bigl(\frac{iπ}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{iπ}{2a}\,Δ_{k,a}\Bigr), \] which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator $Δ_{k,a}=|x|^{2-a}Δ_{k}-|x|^{a}$, $a>0$, where $Δ_{k}$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of $\mathcal{F}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{λ,a}$. We obtain necessary and sufficient cond…

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 $(k,1)$-generali…

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 expans…

具有二十面体对称性的量子谐振子及若干显式波函数

The quantum harmonic oscillator with icosahedral symmetry and some explicit wavefunctions

本文研究H3型正二十面体对称群下的修正量子谐振子，利用Dunkl拉普拉斯算子定义哈密顿量，构造两类特殊波函数并显式计算其平方范数，得到对称不变多项式，还发现了与哈密顿量对易的六阶算子。

二次 Fourier 变换的 Hausdorff--Young 定理的模拟形式及振荡积分算子的有界性

Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator

本文基于 Riesz--Thorin 插值定理证明了二次 Fourier 变换和线性正则变换的 Hausdorff--Young 型不等式，并研究了多项式相位振荡积分算子的有界性。

Regularity theory for the fractional harmonic oscillator

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator $H^σ=(-Δ+|x|^2)^σ$, $0<σ<1$. More precisely, a new class of smooth functions $C^{k,α}_H$ is defined, in which we study the action of $H^σ$. It turns out that these spaces are the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite-Riesz transforms with the Hölder spaces $C^{k,α}_H$ is needed, that we believe of independent interest. The parallel results for the fractional powers of the Laplacian $(-Δ)^σ$ were applied by Caf…