Extension and tangential CRF conditions in quaternionic analysis

The tangential $k$-Cauchy-Fueter complexes and Hartogs' phenomenon over the right quaternionic Heisenberg group

We construct the tangential $k$-Cauchy-Fueter complexes on the right quaternionic Heisenberg group, as the quaternionic counterpart of $\overline{\partial}_b$-complex on the Heisenberg group in the theory of several complex variables. We can use the $L^2$ estimate to solve the nonhomogeneous tangential $k$-Cauchy-Fueter equation under the compatibility condition over this group modulo a lattice. This solution has an important vanishing property when the group is higher dimensional. It allows us to prove the Hartogs' extension phenomenon for $k$-CF functions, which are the quaternionic counterp…

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…

Quaternionic regularity and the dibar-Neumann problem in C^2

Let D be a domain in the quaternionic space H. We prove a differential criterion that characterizes Fueter-regular quaternionic functions f:bD -> H of class C^1. We find differential operators T and N, with complex coefficients, such that a function f is regular on D if and only if (N-jT)f=0 on the boundary of D (j a basic quaternion) and f is harmonic on D. As a consequence, by means of the identification of H with C^2, we obtain a non-tangential holomorphicity condition which generalizes a result of Aronov and Kytmanov. We also show how the differential criterion and regularity are related t…

四元数单位球上的四元数默比乌斯不变拉普拉斯算子与四元数默比乌斯调和函数

Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball

该研究在四元数单位球上构造QM变换与QM不变拉普拉斯算子，求解其狄利克雷问题并证明法图型定理，克服四元数代数非交换性等困难，拓展了调和分析在四元数域的相关理论。

On the boundary complex of the $k$-Cauchy-Fueter complex

The $k$-Cauchy-Fueter complex, $k=0,1,\ldots$, in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy-Riemann complex in complex analysis.They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs-Bochner extension for $k$-regular functions, the quaternionic counterpart of holomorphic fu…

多向量变量的单演函数的Hartogs-Bochner延拓与Dirac复形

The Hartogs-Bochner extension for monogenic functions of several vector variables and the Dirac complex

本文研究多向量变量的单演函数，通过刻画Dirac复形的前四个空间并显式写出前三个算子，证明其为椭圆复形，进而用PDE方法获得非齐次方程解，建立Hartogs现象及边界上切向单演函数的Hartogs-Bochner延拓。