arXiv subjects

Zhongkai Li

Publications and source records attributed to Zhongkai Li.

6 recordsLinked to original sources

arXiv2026

A Bloch type space associated with λ-analytic functions

For $λ\ge0$, the so-called $λ$-analytic functions are defined in terms of the (complex) Dunkl operators $D_{z}$ and $D_{\bar{z}}$. In the paper we introduce a Bloch type space on the disk ${\mathbb D}$ associated with $λ$-analytic functions, called the $λ$-Bloch space and denoted by ${\mathfrak{B}}_λ({\mathbb D})$. Various properties of the $λ$-Bloch space ${\mathfrak{B}}_λ({\mathbb D})$ are proved. We give a characterization of functions in ${\mathfrak{B}}_λ({\mathbb D})$ by means of the higher-order operators $(D_z\circ z)^n$ for $n\ge2$. A general integral operator is proved to be bounded from $L^{\infty}({\mathbb D})$ onto ${\mathfrak{B}}_λ({\mathbb D})$, and as an application, the dual relation of ${\mathfrak{B}}_λ({\mathbb D})$ and the $λ$-Bergman space ($p=1$) is verified.

math.CV↗

arXiv2023

Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions

For $λ\ge0$, a $C^2$ function $f$ defined on the unit disk ${\mathbb D}$ is said to be $λ$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f=\partial_{\bar{z}}f-λ(f(z)-f(\bar{z}))/(z-\bar{z})$. The aim of the paper is to study several problems on the associated Bergman spaces $A^{p}_λ({\mathbb D})$ and Hardy spaces $H_λ^p({\mathbb D})$ for $p\ge2λ/(2λ+1)$, such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of $A^{p}_λ({\mathbb D})$ and $H_λ^p({\mathbb D})$, and characterization and interpolation of $A^{p}_λ({\mathbb D})$.

math.CV↗

arXiv2022

The dual of the Hardy space associated with the Dunkl operators

The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $R^d$ associated with a root system, that contain some non-local refection terms, and the associated Hardy space is defined by means of the Riesz transforms with respect to the Dunkl operators. The aim of the paper is to prove that its dual can be realized by a class of functions on $R^d$, denoted by $BMC_κ$ in the text, that consists of the underlying functions of a certain type of weighted Carleson measures. Our method is "purely analytic" and does not depend on the atomic decomposition. As a corollary we obtain the Fefferman-Stein decomposition of functions in $BMC_κ$.

math.FA↗

arXiv2022

Boundedness of operators on the Bergman spaces associated with a class of generalized analytic functions

The purpose of the paper is to study the operators on the weighted Bergman spaces on the unit disk ${\mathbb{D}}$, denoted by $A^{p}_{λ,w}({\mathbb{D}})$, that are associated with a class of generalized analytic functions, named the $λ$-analytic functions, and with a class of radial weight functions $w$. For $λ\ge0$, a $C^2$ function $f$ on ${\mathbb D}$ is said to be $λ$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f=\partial_{\bar{z}}f-λ(f(z)-f(\bar{z}))/(z-\bar{z})$. It is shown that, for $2λ/(2λ+1)\le p\le1$, the boundedness of an operator from $A^{p}_{λ,w}({\mathbb{D}})$ into a Banach space depends only upon the norm estimate of a single vector-valued $λ$-analytic function. As applications, we obtain a necessary and sufficient conditions of sequence multipliers on the spaces $A^{p}_{λ,w}({\mathbb{D}})$ for general weights $w$, and characterize the dual space of $A^{p}_{λ,w}({\mathbb{D}})$ for the power weight $w=(1-|z|^2)^{α-1}$ with $α>0$, and also give a sufficient condition of Carleson type for boundedness of multiplication operators on $A^{p}_{λ,w}({\mathbb{D}})$.

math.CV↗

arXiv2022

Local boundary behaviour and the area integral of generalized harmonic functions associated with root systems

The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $\RR^d$ associated with a root system. The aim of the paper is to study local boundary behaviour of generalized harmonic functions associated with the Dunkl operators. We introduce a Lusin-type area integral operator $S$ by means of Dunkl's generalized translation and the Dunkl operators. The main results are on characterizations of local existence of non-tangential boundary limits of a generalized harmonic function $u$ in the upper half-space $\RR^{d+1}_+$ associated with the Dunkl operators, and for a subset $E$ of $\RR^d$ invariant under the reflection group generated by the root system, the equivalence of the following three assertions are proved: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)$ is finite for a.e. $x\in E$.

math.FA↗

arXiv2009

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.

math.CA↗