Functions of several quaternion variables and quaternion manifolds

A direct approach to quaternionic manifolds

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on $\mathbb{H}^n$, in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.

Algebraic Regularity over Quaternions and Regular Four-Manifolds

Based on a new generalization of Cauchy-Riemann system presented in this paper, we introduce a class of quaternion-valued functions of a quaternionic variable, which are called algebraic regular functions. The set of algebraic regular functions is not only a real associative algebra, but also respect the composition of functions. Using algebraic regular functions as transition maps, we introduce a class of four-manifolds called the regular four-manifolds.

四元数与八元数上的切片共形性与黎曼流形

Slice conformality and Riemann manifolds on quaternions and octonions

该研究借鉴黎曼共形性思路，定义四元数与八元数黎曼流形的切片共形参数化，涵盖多类典型流形，并给出四元数、八元数对数与n次根函数的统一定义。

From Hankel operators to Carleson measures in a quaternionic variable

We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari anc C. Fefferman are proved.

Differentiable functions of quaternion variables

We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of a quaternion variable to be analytic. In particular, the quaternionic exponential and logarithmic functions are being considered. Main results include quaternion versions of Hurwitz' theorem, Mittag-Leffler's theorem and Weierstrass theorem.

Quaternionic slice regular functions and quaternionic Laplace transforms

The functions studied in the paper are quaternion-valued functions of a quaternionic variable. It is show that the left slice regular functions and right slice regular functions are related by a particular involution. The relation between left slice regular functions, right slice regular functions and intrinsic regular functions is revealed. The classical Laplace transform can be naturally generalized to quaternions in two different ways, which transform a quaternion-valued function of a real variable to a left or right slice regular quaternion-valued function of a quaternionic variable. The u…