四元数Clifford分析的基础II：方程的分裂

The Cauchy transform in the slice hyperholomorphic setting and related topics

In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs.

Weak solutions to the quaternionic Monge-Ampère equation

We solve the Dirichlet problem for the quaternionic Monge-Ampère equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is two which turns out to be less than an integrability exponent of the fundamental solution.

The $\mathcal{F}$-resolvent equation and Riesz projectors for the $\mathcal{F}$-functional calculus

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collabo…

Higher order gradients of monogenic functions

Given a monogenic function on the quaternionic algebra $\mathbb{H}$, the Clifford algebra $\mathbb{R}_n$ or the octonionic algebra $\mathbb{O}$ we prove that $|\nabla^m f|^α$ is subharmonic for some $α>0$ where $\nabla^m f$ is the $m$-th order gradient of $f$. We find also the optimal value of $α$. This is generalization of a result of Calderon and Zygmund.

The quaternionic Monge-Ampère operator and plurisubharmonic functions on the Heisenberg group

Many fundamental results of pluripotential theory on the quaternionic space $\mathbb{H}^n$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge-Ampère operator, differential operators $d_0$ and $d_1$ and a closed positive current on the Heisenberg group. The quaternionic Monge-Ampère operator is the coefficient of $ (d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternioni…

某些全局算子与材料导数

Some global operators and the material derivative

本文探讨了全局算子G的理论及其与切片全纯函数理论的关系，扩展了Clifford分析结构和四元数分析结构的研究，提出了材料导数的性质。

四元数和八元数正则函数的零点：除法引理和凸轮轴效应

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

详细研究了四元数或八元数变量的正则函数的零点集。利用收敛幂级数的一个除法引理，得到了两个八元数正则函数的零点与其乘积的零点之间的精确关系。在八元数多项式的情形下，得到了代数代数基本定理的一个强形式。我们证明了零点的重数之和等于多项式的次数，并得到了线性多项式的一个因式分解。

Douglas-Ahlfors 恒等式的高维版本

Higher Dimensional Versions of the Douglas-Ahlfors Identities

本文研究 Douglas-Ahlfors 恒等式在高维球体中的推广，探讨了欧氏空间调和函数、四元正则函数及 Clifford 单演函数设定下的等价关系，发现 Clifford 代数设定下需用不同规则替代原关系式。

切片单演函数的增长与偏差定理

The Growth and Distortion Theorems for Slice Monogenic Functions

利用一个新的凸组合恒等式，在Clifford代数的背景下，建立了单位圆盘$\mathbb D\subset \mathbb C$上单叶函数片单演扩张的锐增长和偏差定理.类似的结果在片正则函数的四元数情况下也是有效的，在这种情况下我们甚至可以证明Koebe型四分之一定理。我们的增长和扭曲定理切片定期（切片单演）的扩展到更高的维度的单叶全纯函数举行没有额外的几何假设，在几个复变数的设置，其中的增长和扭曲定理失败，一般只适用于一些子类与星形或凸假设。

实替代*代数上的单生成函数：基本结果与应用

Monogenic functions over real alternative *-algebras: fundamental results and applications

本文研究实替代*代数上单生成函数的基本性质，探讨Cauchy-Pompeiu积分公式和超复子空间中的泰勒级数展开，揭示代数结构与超复分析的相互作用。

实替代*代数上的单生成函数：多超复变量情况

Monogenic functions over real alternative *-algebras: the several hypercomplex variables case

本文研究实替代*代数上多超复变量的单生成函数，扩展了多复变量理论至更一般框架，发展了Bochner-Martinelli公式、Plemelj-Sokhotski公式和Hartogs扩展定理等基本性质。

Menelaus定理、Clifford组态和Schwarzian KP族的反演几何

Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

本文证明了可积离散Schwarzian KP（dSKP）方程是一个代数叠加公式，它与Schwarzian KP族、经典达布变换和拟共形映射等有关，但它只包含了古希腊几何的一个基本定理.因此，它表明，与Menelaus定理的连接，更一般地说，Clifford配置呈现dSKP方程的一个自然的对象的反演几何的平面上。讨论了dSKP格的几何可积性和代数可积性及其到Menelaus-Darboux型格、Schwarzian KdV型格、Schwarzian Boussinesq型格和Schramm型格的约化. dSKP和离散Schwarzian Boussinesq方程表示拟共形映射族的离散化。