SUMMARYIn this paper, we consider functions deÿned in a star-like domain ⊂ R n with values in the Cli ord algebra C' 0; n which are polymonogenic with respect to the (left) Dirac operator D = n j=1 e j @=@x j , i.e. they belong to the kernel of D k . We prove that any polymonogenic function f has a decomposition of the formwhere x = x 1 e 1 + · · · + xnen and f j ; j = 1; : : : ; k; are monogenic functions. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition of polynomials. Similar results are obtained for the powers of weighted Dirac operators of the formD = |x| − xD; ∈ R\{0}.

Using the Clifford algebra formalism we study the Möbius gyrogroup of the ball of radius t of the paravector space R ⊕ V , where V is a finite-dimensional real vector space. We characterize all the gyro-subgroups of the Möbius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyrosubgroup for the paravector ball. The geometric and algebraic properties of the equivalence classes are investigated. We show that the equivalence classes locate in a k-dimensional sphere, where k is the dimension of the gyro-subgroup, and the resulting quotient spaces are again Möbius gyrogroups. With the algebraic structure of the factorizations we study the sections of Möbius fiber bundles inherited by the Möbius projectors.

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of f (q) = Σ n∈N (q − p) * n an, given by a σ-ball Σ(p, r), is not open in H unless p ∈ R. This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along this line, we can establish an extension theorem and a representation formula in a slice-domain.

Abstract. The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc D ⊂ C in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.

A slice theory of several octonionic variables is introduced in the article as a generalization of the holomorphic theory of several complex variables. Our new trick is to focus our attentions on some subset of O n with the same complex structures. In our setting, the Bochner-Martinelli formula and the Hartogs extension theorems are established for slice functions of several octonionic variables. KEYWORDS Bochner-Martinelli formula, Hartogs phenomena, octonions, slice regular functions MSC CLASSIFICATION 30G35; 31B10Math Meth Appl Sci. 2020;43:6031-6042.wileyonlinelibrary.com/journal/mma

Abstract. In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.

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The characterization by weighted Lipschitz continuity is given for the Bloch space on the unit ball of R n . Similar results are obtained for little Bloch and Besov spaces.